Modeling and Control of the tPVTOL

ScienceDirect IFAC-PapersOnLine 48-9 (2015) 150–155 Available online at www.sciencedirect.com 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2015.08.075 J.G.Villagómez et al. / IFAC-PapersOnLine 48-9 (2015) 150–155 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Modeling and Control of the tPVTOL  J.G.Villag´omez, M.Vargas, M.G.Ortega, F.R.Rubio Dpt. Systems Engineering and Automation. Univ. of Seville, Spain {villagomez,vargas,mortega,rubio}@us.es Abstract: This paper presents the modeling and control of a planar multibody aerial platform, composed of a Miniature Vertical Take-Off and Landing Aerial Platform (MAV-VTOL) and a camera positioning system. The goal is to improve the current operational profile of visual sensors onboard the MAV, by broadening current aerial configurations for visual sensors tasks with novel capabilities to disengage the dynamic coupling due to typical setups. The Newton- Euler and Euler-Lagrange formalisms are simultaneously applied aiming at obtain and verify coupling terms between the aerial and the camera positioning system. Results of linear and nonlinear control techniques applied to control the position and orientation of the camera frame onboard the MAV are presented. Keywords: Multi-body mechanics, Fly-the-camera, Multi-rotor nonlinear control, Aerial manipulation. 1. INTRODUCTION The development and application of Miniature Air Vehicles (MAV) provides an excellent alternative to typical aerial platforms due to its operational functionalities, such as vertical take-off and landing (VTOL) or maneuverability and hovering, to name a few. The range of electromechanical devices to be attached to the frame, in order to enhance its current functional profile is quite constrained, mainly due to the payload inherent limitations. Works where the modeling of a system compounded by an industrial manipulator and an aerial platform have been recently presented (Kondak et al. (2014)). Whereas this concept is quite promising, with applications such as aerial grasping or cargo transportation, represents new challenges in control theory (Kim et al. (2013)). It is possible to consider other MAV applications, where it would be necessary, e.g., the use of an actuated optical sensor onboard the MAV (Altug et al. (2002)). By combining the 3 translational DOF plus the yaw angle provided by the motion of a MAV with the 2 rotational DOF supplied by a camera positioner, or camera gimbal, it is possible to position the six degrees of freedom of the onboard camera. Previous works already considered this alternative, known as “fly-the-camera”, but only to improve the camera operator experience (Neff et al. (2007)). In contrast, what we propose is the full automation of the compounded system. As a first approach to the task of modeling and control, the model under consideration is a planar assumption of the full roll-tilt VTOL multi-body system, hereby stated as tilt-PVTOL. All references will be given in the camera frame (Villagomez et al. (2014a)), and corresponding dynamic model is hereby obtained. Other considerations are also valid, such as considering the This work was partially supported by the spanish Ministry of Education (MECD) under national research projects DPI2012 − 37580 − C02 − 02 and DPI2013 − 44135 − R. camera evolving and references given in the MAV frame (Villagomez et al. (2014b)). The outline of this work is organized as follows: section II describes the procedure followed to compute the dynamic model of a camera moving in the space through the constrained motion in a plane of a MAV with a camera positioner. Linear and nonlinear control techniques are respectively proposed to stabilize the flight of the camera in sections III and IV. Numerical results are presented in Section V to validate the design of the nonlinear controllers, and final remarks and perspectives are given in section VI. 2. SYSTEM MODELING For modeling the mechanics of the system under consideration we introduce a variation of the well-known Planar Vertical Take-Off and Landing (PVTOL) aircraft problem (Castillo et al. (2005)), with the addition of a camera positioner in configuration roll−tilt, in the vertical plane. This model will be called tPVTOL (tilt PVTOL). Consider a 1 DOF camera positioner system evolving within the plane, thanks to the flight of a miniature rotorcraft, which it is rigidly mounted to. The camera actuator is intended to keep a desired camera inertial orientation, during near-hovering or in-motion maneuvers rejecting disturbances. The in-flight camera positioning operation shifts the rotorcraft center of gravity (CoG), generating reaction forces and consequently disturbing the inertial camera orientation (Orsag et al. (2013)). 2.1 Reference Frames and Notation The kinematics of the flying system comprises three right-handed reference coordinate frames. Let (ˆ xW,0,ˆ zW), which defines the fixed inertial frame W, for the planar case, whose origin OWis located at the Earth surface. Let: (ˆ xB,0,ˆ zB) be the body-fixed frame Bfor the planar case, whose origin OB represents the MAV’s center of masses Proceedings of the 2015 IFAC Workshop on Advanced Control and Navigation for Autonomous Aerospace Vehicles June 10-12, 2015. Seville, Spain Copyright © 2015 IFAC 150 Modeling and Control of the tPVTOL  J.G.Villag´omez, M.Vargas, M.G.Ortega, F.R.Rubio Dpt. Systems Engineering and Automation. Univ. of Seville, Spain {villagomez,vargas,mortega,rubio}@us.es Abstract: This paper presents the modeling and control of a planar multibody aerial platform, composed of a Miniature Vertical Take-Off and Landing Aerial Platform (MAV-VTOL) and a camera positioning system. The goal is to improve the current operational profile of visual sensors onboard the MAV, by broadening current aerial configurations for visual sensors tasks with novel capabilities to disengage the dynamic coupling due to typical setups. The Newton- Euler and Euler-Lagrange formalisms are simultaneously applied aiming at obtain and verify coupling terms between the aerial and the camera positioning system. Results of linear and nonlinear control techniques applied to control the position and orientation of the camera frame onboard the MAV are presented. Keywords: Multi-body mechanics, Fly-the-camera, Multi-rotor nonlinear control, Aerial manipulation. 1. INTRODUCTION The development and application of Miniature Air Vehicles (MAV) provides an excellent alternative to typical aerial platforms due to its operational functionalities, such as vertical take-off and landing (VTOL) or maneuverability and hovering, to name a few. The range of electromechanical devices to be attached to the frame, in order to enhance its current functional profile is quite constrained, mainly due to the payload inherent limitations. Works where the modeling of a system compounded by an industrial manipulator and an aerial platform have been recently presented (Kondak et al. (2014)). Whereas this concept is quite promising, with applications such as aerial grasping or cargo transportation, represents new challenges in control theory (Kim et al. (2013)). It is possible to consider other MAV applications, where it would be necessary, e.g., the use of an actuated optical sensor onboard the MAV (Altug et al. (2002)). By combining the 3 translational DOF plus the yaw angle provided by the motion of a MAV with the 2 rotational DOF supplied by a camera positioner, or camera gimbal, it is possible to position the six degrees of freedom of the onboard camera. Previous works already considered this alternative, known as “fly-the-camera”, but only to improve the camera operator experience (Neff et al. (2007)). In contrast, what we propose is the full automation of the compounded system. As a first approach to the task of modeling and control, the model under consideration is a planar assumption of the full roll-tilt VTOL multi-body system, hereby stated as tilt-PVTOL. All references will be given in the camera frame (Villagomez et al. (2014a)), and corresponding dynamic model is hereby obtained. Other considerations are also valid, such as considering the This work was partially supported by the spanish Ministry of Education (MECD) under national research projects DPI2012 − 37580 −C02 −02 and DPI2013 −44135 −R. camera evolving and references given in the MAV frame (Villagomez et al. (2014b)). The outline of this work is organized as follows: section II describes the procedure followed to compute the dynamic model of a camera moving in the space through the constrained motion in a plane of a MAV with a camera positioner. Linear and nonlinear control techniques are respectively proposed to stabilize the flight of the camera in sections III and IV. Numerical results are presented in Section V to validate the design of the nonlinear controllers, and final remarks and perspectives are given in section VI. 2. SYSTEM MODELING For modeling the mechanics of the system under consideration we introduce a variation of the well-known Planar Vertical Take-Off and Landing (PVTOL) aircraft problem (Castillo et al. (2005)), with the addition of a camera positioner in configuration roll−tilt, in the vertical plane. This model will be called tPVTOL (tilt PVTOL). Consider a 1 DOF camera positioner system evolving within the plane, thanks to the flight of a miniature rotorcraft, which it is rigidly mounted to. The camera actuator is intended to keep a desired camera inertial orientation, during near-hovering or in-motion maneuvers rejecting disturbances. The in-flight camera positioning operation shifts the rotorcraft center of gravity (CoG), generating reaction forces and consequently disturbing the inertial camera orientation (Orsag et al. (2013)). 2.1 Reference Frames and Notation The kinematics of the flying system comprises three right-handed reference coordinate frames. Let (ˆ xW,0,ˆ zW), which defines the fixed inertial frame W, for the planar case, whose origin OWis located at the Earth surface. Let: (ˆ xB,0,ˆ zB) be the body-fixed frame Bfor the planar case, whose origin OBrepresents the MAV’s center of masses Proceedings of the 2015 IFAC Workshop on Advanced Control and Navigation for Autonomous Aerospace Vehicles June 10-12, 2015. Seville, Spain Copyright © 2015 IFAC 150 Modeling and Control of the tPVTOL  J.G.Villag´omez, M.Vargas, M.G.Ortega, F.R.Rubio Dpt. Systems Engineering and Automation. Univ. of Seville, Spain {villagomez,vargas,mortega,rubio}@us.es Abstract: This paper presents the modeling and control of a planar multibody aerial platform, composed of a Miniature Vertical Take-Off and Landing Aerial Platform (MAV-VTOL) and a camera positioning system. The goal is to improve the current operational profile of visual sensors onboard the MAV, by broadening current aerial configurations for visual sensors tasks with novel capabilities to disengage the dynamic coupling due to typical setups. The Newton- Euler and Euler-Lagrange formalisms are simultaneously applied aiming at obtain and verify coupling terms between the aerial and the camera positioning system. Results of linear and nonlinear control techniques applied to control the position and orientation of the camera frame onboard the MAV are presented. Keywords: Multi-body mechanics, Fly-the-camera, Multi-rotor nonlinear control, Aerial manipulation. 1. INTRODUCTION The development and application of Miniature Air Vehicles (MAV) provides an excellent alternative to typical aerial platforms due to its operational functionalities, such as vertical take-off and landing (VTOL) or maneuverability and hovering, to name a few. The range of electromechanical devices to be attached to the frame, in order to enhance its current functional profile is quite constrained, mainly due to the payload inherent limitations. Works where the modeling of a system compounded by an industrial manipulator and an aerial platform have been recently presented (Kondak et al. (2014)). Whereas this concept is quite promising, with applications such as aerial grasping or cargo transportation, represents new challenges in control theory (Kim et al. (2013)). It is possible to consider other MAV applications, where it would be necessary, e.g., the use of an actuated optical sensor onboard the MAV (Altug et al. (2002)). By combining the 3 translational DOF plus the yaw angle provided by the motion of a MAV with the 2 rotational DOF supplied by a camera positioner, or camera gimbal, it is possible to position the six degrees of freedom of the onboard camera. Previous works already considered this alternative, known as “fly-the-camera”, but only to improve the camera operator experience (Neff et al. (2007)). In contrast, what we propose is the full automation of the compounded system. As a first approach to the task of modeling and control, the model under consideration is a planar assumption of the full roll-tilt VTOL multi-body system, hereby stated as tilt-PVTOL. All references will be given in the camera frame (Villagomez et al. (2014a)), and corresponding dynamic model is hereby obtained. Other considerations are also valid, such as considering the This work was partially supported by the spanish Ministry of Education (MECD) under national research projects DPI2012 − 37580 −C02 −02 and DPI2013 −44135 −R. camera evolving and references given in the MAV frame (Villagomez et al. (2014b)). The outline of this work is organized as follows: section II describes the procedure followed to compute the dynamic model of a camera moving in the space through the constrained motion in a plane of a MAV with a camera positioner. Linear and nonlinear control techniques are respectively proposed to stabilize the flight of the camera in sections III and IV. Numerical results are presented in Section V to validate the design of the nonlinear controllers, and final remarks and perspectives are given in section VI. 2. SYSTEM MODELING For modeling the mechanics of the system under consideration we introduce a variation of the well-known Planar Vertical Take-Off and Landing (PVTOL) aircraft problem (Castillo et al. (2005)), with the addition of a camera positioner in configuration roll−tilt, in the vertical plane. This model will be called tPVTOL (tilt PVTOL). Consider a 1 DOF camera positioner system evolving within the plane, thanks to the flight of a miniature rotorcraft, which it is rigidly mounted to. The camera actuator is intended to keep a desired camera inertial orientation, during near-hovering or in-motion maneuvers rejecting disturbances. The in-flight camera positioning operation shifts the rotorcraft center of gravity (CoG), generating reaction forces and consequently disturbing the inertial camera orientation (Orsag et al. (2013)). 2.1 Reference Frames and Notation The kinematics of the flying system comprises three right-handed reference coordinate frames. Let (ˆ xW,0,ˆ zW), which defines the fixed inertial frame W, for the planar case, whose origin OWis located at the Earth surface. Let: (ˆ xB,0,ˆ zB) be the body-fixed frame Bfor the planar case, whose origin OBrepresents the MAV’s center of masses Proceedings of the 2015 IFAC Workshop on Advanced Control and Navigation for Autonomous Aerospace Vehicles June 10-12, 2015. Seville, Spain Copyright © 2015 IFAC 150 Modeling and Control of the tPVTOL  J.G.Villag´omez, M.Vargas, M.G.Ortega, F.R.Rubio Dpt. Systems Engineering and Automation. Univ. of Seville, Spain {villagomez,vargas,mortega,rubio}@us.es Abstract: This paper presents the modeling and control of a planar multibody aerial platform, composed of a Miniature Vertical Take-Off and Landing Aerial Platform (MAV-VTOL) and a camera positioning system. The goal is to improve the current operational profile of visual sensors onboard the MAV, by broadening current aerial configurations for visual sensors tasks with novel capabilities to disengage the dynamic coupling due to typical setups. The Newton- Euler and Euler-Lagrange formalisms are simultaneously applied aiming at obtain and verify coupling terms between the aerial and the camera positioning system. Results of linear and nonlinear control techniques applied to control the position and orientation of the camera frame onboard the MAV are presented. Keywords: Multi-body mechanics, Fly-the-camera, Multi-rotor nonlinear control, Aerial manipulation. 1. INTRODUCTION The development and application of Miniature Air Vehicles (MAV) provides an excellent alternative to typical aerial platforms due to its operational functionalities, such as vertical take-off and landing (VTOL) or maneuverability and hovering, to name a few. The range of electromechanical devices to be attached to the frame, in order to enhance its current functional profile is quite constrained, mainly due to the payload inherent limitations. Works where the modeling of a system compounded by an industrial manipulator and an aerial platform have been recently presented (Kondak et al. (2014)). Whereas this concept is quite promising, with applications such as aerial grasping or cargo transportation, represents new challenges in control theory (Kim et al. (2013)). It is possible to consider other MAV applications, where it would be necessary, e.g., the use of an actuated optical sensor onboard the MAV (Altug et al. (2002)). By combining the 3 translational DOF plus the yaw angle provided by the motion of a MAV with the 2 rotational DOF supplied by a camera positioner, or camera gimbal, it is possible to position the six degrees of freedom of the onboard camera. Previous works already considered this alternative, known as “fly-the-camera”, but only to improve the camera operator experience (Neff et al. (2007)). In contrast, what we propose is the full automation of the compounded system. As a first approach to the task of modeling and control, the model under consideration is a planar assumption of the full roll-tilt VTOL multi-body system, hereby stated as tilt-PVTOL. All references will be given in the camera frame (Villagomez et al. (2014a)), and corresponding dynamic model is hereby obtained. Other considerations are also valid, such as considering the This work was partially supported by the spanish Ministry of Education (MECD) under national research projects DPI2012 − 37580 −C02 −02 and DPI2013 −44135 −R. camera evolving and references given in the MAV frame (Villagomez et al. (2014b)). The outline of this work is organized as follows: section II describes the procedure followed to compute the dynamic model of a camera moving in the space through the constrained motion in a plane of a MAV with a camera positioner. Linear and nonlinear control techniques are respectively proposed to stabilize the flight of the camera in sections III and IV. Numerical results are presented in Section V to validate the design of the nonlinear controllers, and final remarks and perspectives are given in section VI. 2. SYSTEM MODELING For modeling the mechanics of the system under consideration we introduce a variation of the well-known Planar Vertical Take-Off and Landing (PVTOL) aircraft problem (Castillo et al. (2005)), with the addition of a camera positioner in configuration roll−tilt, in the vertical plane. This model will be called tPVTOL (tilt PVTOL). Consider a 1 DOF camera positioner system evolving within the plane, thanks to the flight of a miniature rotorcraft, which it is rigidly mounted to. The camera actuator is intended to keep a desired camera inertial orientation, during near-hovering or in-motion maneuvers rejecting disturbances. The in-flight camera positioning operation shifts the rotorcraft center of gravity (CoG), generating reaction forces and consequently disturbing the inertial camera orientation (Orsag et al. (2013)). 2.1 Reference Frames and Notation The kinematics of the flying system comprises three right-handed reference coordinate frames. Let (ˆ xW,0,ˆ zW), which defines the fixed inertial frame W, for the planar case, whose origin OWis located at the Earth surface. Let: (ˆ xB,0,ˆ zB) be the body-fixed frame Bfor the planar case, whose origin OBrepresents the MAV’s center of masses Proceedings of the 2015 IFAC Workshop on Advanced Control and Navigation for Autonomous Aerospace Vehicles June 10-12, 2015. Seville, Spain Copyright © 2015 IFAC 150 J.G.Villagómez et al. / IFAC-PapersOnLine 48-9 (2015) 150–155 151 Modeling and Control of the tPVTOL  J.G.Villag´omez, M.Vargas, M.G.Ortega, F.R.Rubio Dpt. Systems Engineering and Automation. Univ. of Seville, Spain {villagomez,vargas,mortega,rubio}@us.es Abstract: This paper presents the modeling and control of a planar multibody aerial platform, composed of a Miniature Vertical Take-Off and Landing Aerial Platform (MAV-VTOL) and a camera positioning system. The goal is to improve the current operational profile of visual sensors onboard the MAV, by broadening current aerial configurations for visual sensors tasks with novel capabilities to disengage the dynamic coupling due to typical setups. The Newton- Euler and Euler-Lagrange formalisms are simultaneously applied aiming at obtain and verify coupling terms between the aerial and the camera positioning system. Results of linear and nonlinear control techniques applied to control the position and orientation of the camera frame onboard the MAV are presented. Keywords: Multi-body mechanics, Fly-the-camera, Multi-rotor nonlinear control, Aerial manipulation. 1. INTRODUCTION The development and application of Miniature Air Vehicles (MAV) provides an excellent alternative to typical aerial platforms due to its operational functionalities, such as vertical take-off and landing (VTOL) or maneuverability and hovering, to name a few. The range of electromechanical devices to be attached to the frame, in order to enhance its current functional profile is quite constrained, mainly due to the payload inherent limitations. Works where the modeling of a system compounded by an industrial manipulator and an aerial platform have been recently presented (Kondak et al. (2014)). Whereas this concept is quite promising, with applications such as aerial grasping or cargo transportation, represents new challenges in control theory (Kim et al. (2013)). It is possible to consider other MAV applications, where it would be necessary, e.g., the use of an actuated optical sensor onboard the MAV (Altug et al. (2002)). By combining the 3 translational DOF plus the yaw angle provided by the motion of a MAV with the 2 rotational DOF supplied by a camera positioner, or camera gimbal, it is possible to position the six degrees of freedom of the onboard camera. Previous works already considered this alternative, known as “fly-the-camera”, but only to improve the camera operator experience (Neff et al. (2007)). In contrast, what we propose is the full automation of the compounded system. As a first approach to the task of modeling and control, the model under consideration is a planar assumption of the full roll-tilt VTOL multi-body system, hereby stated as tilt-PVTOL. All references will be given in the camera frame (Villagomez et al. (2014a)), and corresponding dynamic model is hereby obtained. Other considerations are also valid, such as considering the This work was partially supported by the spanish Ministry of Education (MECD) under national research projects DPI2012 − 37580 −C02 −02 and DPI2013 −44135 −R. camera evolving and references given in the MAV frame (Villagomez et al. (2014b)). The outline of this work is organized as follows: section II describes the procedure followed to compute the dynamic model of a camera moving in the space through the constrained motion in a plane of a MAV with a camera positioner. Linear and nonlinear control techniques are respectively proposed to stabilize the flight of the camera in sections III and IV. Numerical results are presented in Section V to validate the design of the nonlinear controllers, and final remarks and perspectives are given in section VI. 2. SYSTEM MODELING For modeling the mechanics of the system under consideration we introduce a variation of the well-known Planar Vertical Take-Off and Landing (PVTOL) aircraft problem (Castillo et al. (2005)), with the addition of a camera positioner in configuration roll−tilt, in the vertical plane. This model will be called tPVTOL (tilt PVTOL). Consider a 1 DOF camera positioner system evolving within the plane, thanks to the flight of a miniature rotorcraft, which it is rigidly mounted to. The camera actuator is intended to keep a desired camera inertial orientation, during near-hovering or in-motion maneuvers rejecting disturbances. The in-flight camera positioning operation shifts the rotorcraft center of gravity (CoG), generating reaction forces and consequently disturbing the inertial camera orientation (Orsag et al. (2013)). 2.1 Reference Frames and Notation The kinematics of the flying system comprises three right-handed reference coordinate frames. Let (ˆ xW,0,ˆ zW), which defines the fixed inertial frame W, for the planar case, whose origin OWis located at the Earth surface. Let: (ˆ xB,0,ˆ zB) be the body-fixed frame Bfor the planar case, whose origin OBrepresents the MAV’s center of masses Proceedings of the 2015 IFAC Workshop on Advanced Control and Navigation for Autonomous Aerospace Vehicles June 10-12, 2015. Seville, Spain Copyright © 2015 IFAC 150 Modeling and Control of the tPVTOL  J.G.Villag´omez, M.Vargas, M.G.Ortega, F.R.Rubio Dpt. Systems Engineering and Automation. Univ. of Seville, Spain {villagomez,vargas,mortega,rubio}@us.es Abstract: This paper presents the modeling and control of a planar multibody aerial platform, composed of a Miniature Vertical Take-Off and Landing Aerial Platform (MAV-VTOL) and a camera positioning system. The goal is to improve the current operational profile of visual sensors onboard the MAV, by broadening current aerial configurations for visual sensors tasks with novel capabilities to disengage the dynamic coupling due to typical setups. The Newton- Euler and Euler-Lagrange formalisms are simultaneously applied aiming at obtain and verify coupling terms between the aerial and the camera positioning system. Results of linear and nonlinear control techniques applied to control the position and orientation of the camera frame onboard the MAV are presented. Keywords: Multi-body mechanics, Fly-the-camera, Multi-rotor nonlinear control, Aerial manipulation. 1. INTRODUCTION The development and application of Miniature Air Vehicles (MAV) provides an excellent alternative to typical aerial platforms due to its operational functionalities, such as vertical take-off and landing (VTOL) or maneuverability and hovering, to name a few. The range of electromechanical devices to be attached to the frame, in order to enhance its current functional profile is quite constrained, mainly due to the payload inherent limitations. Works where the modeling of a system compounded by an industrial manipulator and an aerial platform have been recently presented (Kondak et al. (2014)). Whereas this concept is quite promising, with applications such as aerial grasping or cargo transportation, represents new challenges in control theory (Kim et al. (2013)). It is possible to consider other MAV applications, where it would be necessary, e.g., the use of an actuated optical sensor onboard the MAV (Altug et al. (2002)). By combining the 3 translational DOF plus the yaw angle provided by the motion of a MAV with the 2 rotational DOF supplied by a camera positioner, or camera gimbal, it is possible to position the six degrees of freedom of the onboard camera. Previous works already considered this alternative, known as “fly-the-camera”, but only to improve the camera operator experience (Neff et al. (2007)). In contrast, what we propose is the full automation of the compounded system. As a first approach to the task of modeling and control, the model under consideration is a planar assumption of the full roll-tilt VTOL multi-body system, hereby stated as tilt-PVTOL. All references will be given in the camera frame (Villagomez et al. (2014a)), and corresponding dynamic model is hereby obtained. Other considerations are also valid, such as considering the This work was partially supported by the spanish Ministry of Education (MECD) under national research projects DPI2012 − 37580 −C02 −02 and DPI2013 −44135 −R. camera evolving and references given in the MAV frame (Villagomez et al. (2014b)). The outline of this work is organized as follows: section II describes the procedure followed to compute the dynamic model of a camera moving in the space through the constrained motion in a plane of a MAV with a camera positioner. Linear and nonlinear control techniques are respectively proposed to stabilize the flight of the camera in sections III and IV. Numerical results are presented in Section V to validate the design of the nonlinear controllers, and final remarks and perspectives are given in section VI. 2. SYSTEM MODELING For modeling the mechanics of the system under consideration we introduce a variation of the well-known Planar Vertical Take-Off and Landing (PVTOL) aircraft problem (Castillo et al. (2005)), with the addition of a camera positioner in configuration roll−tilt, in the vertical plane. This model will be called tPVTOL (tilt PVTOL). Consider a 1 DOF camera positioner system evolving within the plane, thanks to the flight of a miniature rotorcraft, which it is rigidly mounted to. The camera actuator is intended to keep a desired camera inertial orientation, during near-hovering or in-motion maneuvers rejecting disturbances. The in-flight camera positioning operation shifts the rotorcraft center of gravity (CoG), generating reaction forces and consequently disturbing the inertial camera orientation (Orsag et al. (2013)). 2.1 Reference Frames and Notation The kinematics of the flying system comprises three right-handed reference coordinate frames. Let (ˆ xW,0,ˆ zW), which defines the fixed inertial frame W, for the planar case, whose origin OWis located at the Earth surface. Let: (ˆ xB,0,ˆ zB) be the body-fixed frame Bfor the planar case, whose origin OBrepresents the MAV’s center of masses Proceedings of the 2015 IFAC Workshop on Advanced Control and Navigation for Autonomous Aerospace Vehicles June 10-12, 2015. Seville, Spain Copyright © 2015 IFAC 150 Modeling and Control of the tPVTOL  J.G.Villag´omez, M.Vargas, M.G.Ortega, F.R.Rubio Dpt. Systems Engineering and Automation. Univ. of Seville, Spain {villagomez,vargas,mortega,rubio}@us.es Abstract: This paper presents the modeling and control of a planar multibody aerial platform, composed of a Miniature Vertical Take-Off and Landing Aerial Platform (MAV-VTOL) and a camera positioning system. The goal is to improve the current operational profile of visual sensors onboard the MAV, by broadening current aerial configurations for visual sensors tasks with novel capabilities to disengage the dynamic coupling due to typical setups. The Newton- Euler and Euler-Lagrange formalisms are simultaneously applied aiming at obtain and verify coupling terms between the aerial and the camera positioning system. Results of linear and nonlinear control techniques applied to control the position and orientation of the camera frame onboard the MAV are presented. Keywords: Multi-body mechanics, Fly-the-camera, Multi-rotor nonlinear control, Aerial manipulation. 1. INTRODUCTION The development and application of Miniature Air Vehicles (MAV) provides an excellent alternative to typical aerial platforms due to its operational functionalities, such as vertical take-off and landing (VTOL) or maneuverability and hovering, to name a few. The range of electromechanical devices to be attached to the frame, in order to enhance its current functional profile is quite constrained, mainly due to the payload inherent limitations. Works where the modeling of a system compounded by an industrial manipulator and an aerial platform have been recently presented (Kondak et al. (2014)). Whereas this concept is quite promising, with applications such as aerial grasping or cargo transportation, represents new challenges in control theory (Kim et al. (2013)). It is possible to consider other MAV applications, where it would be necessary, e.g., the use of an actuated optical sensor onboard the MAV (Altug et al. (2002)). By combining the 3 translational DOF plus the yaw angle provided by the motion of a MAV with the 2 rotational DOF supplied by a camera positioner, or camera gimbal, it is possible to position the six degrees of freedom of the onboard camera. Previous works already considered this alternative, known as “fly-the-camera”, but only to improve the camera operator experience (Neff et al. (2007)). In contrast, what we propose is the full automation of the compounded system. As a first approach to the task of modeling and control, the model under consideration is a planar assumption of the full roll-tilt VTOL multi-body system, hereby stated as tilt-PVTOL. All references will be given in the camera frame (Villagomez et al. (2014a)), and corresponding dynamic model is hereby obtained. Other considerations are also valid, such as considering the This work was partially supported by the spanish Ministry of Education (MECD) under national research projects DPI2012 − 37580 −C02 −02 and DPI2013 −44135 −R. camera evolving and references given in the MAV frame (Villagomez et al. (2014b)). The outline of this work is organized as follows: section II describes the procedure followed to compute the dynamic model of a camera moving in the space through the constrained motion in a plane of a MAV with a camera positioner. Linear and nonlinear control techniques are respectively proposed to stabilize the flight of the camera in sections III and IV. Numerical results are presented in Section V to validate the design of the nonlinear controllers, and final remarks and perspectives are given in section VI. 2. SYSTEM MODELING For modeling the mechanics of the system under consideration we introduce a variation of the well-known Planar Vertical Take-Off and Landing (PVTOL) aircraft problem (Castillo et al. (2005)), with the addition of a camera positioner in configuration roll−tilt, in the vertical plane. This model will be called tPVTOL (tilt PVTOL). Consider a 1 DOF camera positioner system evolving within the plane, thanks to the flight of a miniature rotorcraft, which it is rigidly mounted to. The camera actuator is intended to keep a desired camera inertial orientation, during near-hovering or in-motion maneuvers rejecting disturbances. The in-flight camera positioning operation shifts the rotorcraft center of gravity (CoG), generating reaction forces and consequently disturbing the inertial camera orientation (Orsag et al. (2013)). 2.1 Reference Frames and Notation The kinematics of the flying system comprises three right-handed reference coordinate frames. Let (ˆ xW,0,ˆ zW), which defines the fixed inertial frame W, for the planar case, whose origin OWis located at the Earth surface. Let: (ˆ xB,0,ˆ zB) be the body-fixed frame Bfor the planar case, whose origin OBrepresents the MAV’s center of masses Proceedings of the 2015 IFAC Workshop on Advanced Control and Navigation for Autonomous Aerospace Vehicles June 10-12, 2015. Seville, Spain Copyright © 2015 IFAC 150 Modeling and Control of the tPVTOL  J.G.Villag´omez, M.Vargas, M.G.Ortega, F.R.Rubio Dpt. Systems Engineering and Automation. Univ. of Seville, Spain {villagomez,vargas,mortega,rubio}@us.es Abstract: This paper presents the modeling and control of a planar multibody aerial platform, composed of a Miniature Vertical Take-Off and Landing Aerial Platform (MAV-VTOL) and a camera positioning system. The goal is to improve the current operational profile of visual sensors onboard the MAV, by broadening current aerial configurations for visual sensors tasks with novel capabilities to disengage the dynamic coupling due to typical setups. The Newton- Euler and Euler-Lagrange formalisms are simultaneously applied aiming at obtain and verify coupling terms between the aerial and the camera positioning system. Results of linear and nonlinear control techniques applied to control the position and orientation of the camera frame onboard the MAV are presented. Keywords: Multi-body mechanics, Fly-the-camera, Multi-rotor nonlinear control, Aerial manipulation. 1. INTRODUCTION The development and application of Miniature Air Vehicles (MAV) provides an excellent alternative to typical aerial platforms due to its operational functionalities, such as vertical take-off and landing (VTOL) or maneuverability and hovering, to name a few. The range of electromechanical devices to be attached to the frame, in order to enhance its current functional profile is quite constrained, mainly due to the payload inherent limitations. Works where the modeling of a system compounded by an industrial manipulator and an aerial platform have been recently presented (Kondak et al. (2014)). Whereas this concept is quite promising, with applications such as aerial grasping or cargo transportation, represents new challenges in control theory (Kim et al. (2013)). It is possible to consider other MAV applications, where it would be necessary, e.g., the use of an actuated optical sensor onboard the MAV (Altug et al. (2002)). By combining the 3 translational DOF plus the yaw angle provided by the motion of a MAV with the 2 rotational DOF supplied by a camera positioner, or camera gimbal, it is possible to position the six degrees of freedom of the onboard camera. Previous works already considered this alternative, known as “fly-the-camera”, but only to improve the camera operator experience (Neff et al. (2007)). In contrast, what we propose is the full automation of the compounded system. As a first approach to the task of modeling and control, the model under consideration is a planar assumption of the full roll-tilt VTOL multi-body system, hereby stated as tilt-PVTOL. All references will be given in the camera frame (Villagomez et al. (2014a)), and corresponding dynamic model is hereby obtained. Other considerations are also valid, such as considering the This work was partially supported by the spanish Ministry of Education (MECD) under national research projects DPI2012 − 37580 −C02 −02 and DPI2013 −44135 −R. camera evolving and references given in the MAV frame (Villagomez et al. (2014b)). The outline of this work is organized as follows: section II describes the procedure followed to compute the dynamic model of a camera moving in the space through the constrained motion in a plane of a MAV with a camera positioner. Linear and nonlinear control techniques are respectively proposed to stabilize the flight of the camera in sections III and IV. Numerical results are presented in Section V to validate the design of the nonlinear controllers, and final remarks and perspectives are given in section VI. 2. SYSTEM MODELING For modeling the mechanics of the system under consideration we introduce a variation of the well-known Planar Vertical Take-Off and Landing (PVTOL) aircraft problem (Castillo et al. (2005)), with the addition of a camera positioner in configuration roll−tilt, in the vertical plane. This model will be called tPVTOL (tilt PVTOL). Consider a 1 DOF camera positioner system evolving within the plane, thanks to the flight of a miniature rotorcraft, which it is rigidly mounted to. The camera actuator is intended to keep a desired camera inertial orientation, during near-hovering or in-motion maneuvers rejecting disturbances. The in-flight camera positioning operation shifts the rotorcraft center of gravity (CoG), generating reaction forces and consequently disturbing the inertial camera orientation (Orsag et al. (2013)). 2.1 Reference Frames and Notation The kinematics of the flying system comprises three right-handed reference coordinate frames. Let (ˆ xW,0,ˆ zW), which defines the fixed inertial frame W, for the planar case, whose origin OWis located at the Earth surface. Let: (ˆ xB,0,ˆ zB) be the body-fixed frame Bfor the planar case, whose origin OBrepresents the MAV’s center of masses Proceedings of the 2015 IFAC Workshop on Advanced Control and Navigation for Autonomous Aerospace Vehicles June 10-12, 2015. Seville, Spain Copyright © 2015 IFAC 150 Figure 1. Freebody diagram: (a) tPVTOL Frames of reference and (b) depiction (in red) of applied forces and moments. (CMB) location. The camera fixed frame Ccan be denoted as (ˆ xC,0,ˆ zC), whose origin OCcorresponds to the center of the camera frame. The orientation of the rigid body is given by a rotation R:B→W, with R∈SO(2). θB is denoted as the MAV’s inertial pitch angle; θCas the inertial angle of the camera positioner (the camera’s tilt inertial angle) and θCB is the angular difference between the MAV’s pitch angle and the camera’s inertial frame tilt angle. Links 11,12 and 2represents system lengths, as depicted in (Fig. 1). Conceptually four control signals are designed to control the rotational ([τBτCB]) and translational ([fxfz]) subsystems: the control input f=[fxfz]=R(θB)·[0 f]as the thrust vector, where f=f1+f2is the total upward thrust in the body frame, which generates a reaction torque defined as τB=(f2−f1); and the camera positioner rotational joint torque, labelled as τCB.is considered as the length between the position of each propeller and the geometric center of the airframe (symmetric frame). The instantaneous camera inertial position, defined as ξ=[xC,z C]is achieved through the MAV heading’s angle and the total upward thrust, while the camera’s tilt angle is controlled by mixing control signal inputs τBand τCB. 2.2 Newton-Euler Formalism The equations of motion are first computed through the Newton-Euler formulation. The composite system can be first seen as two different bodies subject to external forces, applied to the center of mass of the MAV and the camera frame respectively, expressed in the body-fixed frame. Couplings are obtained according to Newton’s third law. Similarly, forces applied in the MAV frame will be directly passed through the kinematic chain till the camera frame. Consequently, Newton’s equations of motion yield the following dynamic model for que quad-rotor airframe: mB¨xB=fx−Tx mB¨zB=fz+Tz−mBg ICMB˙ωB=τB−τT+Tx12 cos(θB)+Tz12 sin(θB)(1) The projection of the total upward thrust force is stated through components fxand fz.τCB is the applied torque provided by the camera positioner rotational joint, generating reaction forces Txand Tzand a reaction torque labelled as τTin the MAV frame. Excluding the total mass of the assembly comprising the camera and the positioner (henceforth Camera Composite System - CCS), the total mass of the quad-rotor is denoted as mBwith ICMBas the inertia mass-moment. ˙ωBis considered as the angular acceleration of the aircraft expressed in the inertial frame, and g is the gravitational constant. Since the dynamic model is computed from the camera frame perspective, the relationship is stated through the following coordinates xB=xC+12 sin(θB)−2cos(θCB +θB) zB=zC+12 cos(θB)+2sin(θCB +θB) (2) Similarly, the dynamic model of the CCS is stated as follows: mC¨xC=Tx mC¨zC=−Tz−mCg IC˙ωCB =τCB +Txsin(θC)2−Tzcos(θC)2 (3) with τT=τCB, where ICis the inertia mass-moment of the kinetic chain compounded by link 2and camera, with total mass mc, and ˙ωCB is the angular acceleration exerted by the camera frame expressed in the inertial frame. 2.3 Euler-Lagrange Formalism In order to validate the model obtained through the Newton-Euler formalism, the study of the dynamic model by the decomposition of the resulting mechanical energy is proposed. (1) Kinetic Energy The total kinetic energy function Kof the compounded mechanical system resulting from the translational and rotational motion can be partitioned by the sum of the MAV’s kinetic energy, given by KB=1 2mB˙x2 B+˙z2 B+1 2ICMB˙ θB 2(4) and the camera actuator’s kinetic energy, KC=1 2mC(˙x2 C+˙z2 C)+1 2IC(˙ θCB +˙ θB)2(5) where the position of the camera frame center is computed through: xC=xB+2cos(θB+θCB)−12 sin(θB) zC=zB−2sin(θB+θCB)−12 cos(θB) (2) Potential Energy Considering zBthe height of the quad-rotor’s CoG, the potential energy term Uis given by the sum of both aircraft’s and CCS’s potential energies: U=UB+UC=g(mBzB+mCzC) (6) Once the kinetic and potential energy of the compounded system have been computed using (4), (5), and (6), and by definition: L=(KB+KC)−(UB+UC) (7) the Lagrangian L(qL,˙ qL), denoted as LL, can be written as follows: LL=1 2m˙x2 B+ICMB˙ θ2 B+m B˙z2 B+m C˙z2 B+2 12m C˙ θ2 B +m C˙xB−2˙ θBs(θB+θCB)+ ˙ θCBs(θB+θCB)−12 ˙ θBcθB −2m C˙ θB˙zBc(θB+θCB)+ ˙ θCB ˙zBc(θB+θCB) +12m C˙ θB˙zBsθB+2˙ θB˙ θCBsθCB +˙ θ2 BsθCB +2 2m C˙ θ2 B+˙ θ2 CB+2˙ θB˙ θCB +m Cg(2s(θB+θCB)+12cθB)−zBmg (8) IFAC ACNAAV 2015 June 10-12, 2015. Seville, Spain 151 152 J.G.Villagómez et al. / IFAC-PapersOnLine 48-9 (2015) 150–155 by defining the generalized coordinates vector qLas qL=[xBzBθBθCB ](9) The equation (eq. 8) is used for computing the multi-body motion equations, by direct application of the Lagrange formalism:   fx fz τB τCB  =d dt ∂LL ∂˙q L−∂LL ∂qL (10) Since we are interested in the evolution of the compounded system w.r.t. the inertial camera frame position, an additional conversion must be applied to the resulting equations. Hence, by defining the final generalized coordinates vector as q=[xCzCθBθC](11) and rewriting the motion equations by introducing θC=θCB +θB(12) finally, undoing the previous translation between frames according to (Eq. 2), motion equations describing the evolution of the camera frame [xCzCθC]onboard the tPVTOL, are obtained. The total mass of the system is denoted as m=mB+mC, and c(·) and s(·) stands for cos(·) and sin(·), respectively. 2.4 Equations of motion Through the Newton-Euler and Lagrange-Euler formulation, equations modeling the overall motion of the tPV- TOL are derived. Set of equations (2) are used in (1) to get the absolute motion of the camera frame given by (3), considering translational and rotational motion of both systems as two interconnected dynamics (Castillo et al. (2005)). Hence, the translational subsystem evolution is described by the following equations:    fx=m¨xC+mB(12(cθB¨ θB−sθB˙ θB 2)+2(sθC¨ θC+cθC˙ θC 2)) fz=m(¨zC+g)−mB12(sθB¨ θB+cθB˙ θB 2)+mB2(cθC¨ θC−sθC˙ θC 2) (13) and the rotational motion set of equations, which is not directly related with ¨xCand ¨zC, as follows:    τB=τCB +a 1¨ θB+a 3(sθCB ¨ θC+cθCB ˙ θC 2) τCB =a 2¨ θC+a 3(sθCB ¨ θB+cθCB ˙ θB 2)−a4cθCB (14) with: a1=ICMB+mBmC m2 12 a2=IC−mBmC m2 2 a3=mBmC m122a4=m C m2 3. LINEAR CONTROL OF THE TPVTOL In order to evaluate more sophisticated control structures, the study of linear control laws was previously proposed by authors in (Villagomez et al. (2014a)). A classical timescale separation is considered between rotational (fastdynamics inner loop) and translational motion (slow dynamics outer-loop). Both dynamics are partially decoupled and controlled by an attitude and position linear controllers. 3.1 Model asumptions and linearization In hover mode the vehicle will operate in regions where θB0o. It will be used a linearized version of the nonlinear model obtained in (Eqs. 13-14), found by means of Taylor approximation in the vicinity of the nominal operating point. Considering the camera positioner angle around a fixed value (θC0o), the following equilibrium conditions are obtained: θeq B=0 τeq B=τeq CB +m Cg12sθeq B=τeq CB θeq C=0 τeq CB =−m Cg2cθeq C=−m Cg2 feq =mg cθeq B (15) given by the small value increment (small angle assumption) of the generalized coordinates: τB(t)=τeq B+τδ B(t)xC(t)=xeq C+xδ C(t)θB(t)=θeq B+θδ B(t) τCB(t)=τeq CB +τδ CB(t)zC(t)=zeq C+zδ C(t)θC(t)=θeq C+θδ C(t) f(t)=feq +fδ(t) (16) whatever xeq Cand zeq Ccoordinates. This assumption leads to the following time-invariant linearized representation of the system, i.e. the translational: ¨ xδ C(t)=feq mθδ B(t)−m B12 m¨ θδ B(t) (17) ¨ zδ C(t)=fδ(t) m−m B2 m¨ θδ C(t) (18) and the rotational subsystem: τδ B (t)−τδ CB (t)=a 1¨ θδ B(t)+a 3sθeq CB ¨ θδ C(t) (19) τδ CB(t)=a 2˙ θδ C(t)+a 3sθeq CB ¨ θδ B(t) −a4cθeq CBfδ(t)+mCg2sθeq CBθδ CB(t) (20) Based on equations (16-20) in the time domain, the application of Laplace transformation yield the following transfer functions, which are considered for the control tasks: xC(s) = 1 ms2(feq −mB12s2)θB(s) (21) zC(s) = 1 ms2f(s) −m B m2θC(s) (22) θB(s) = 1 a1s2(τB(s) −τCB(s)) (23) θC(s) = 1 a2s2(τCB(s)+a 4f(s)) (24) where, e.g., xC(s) = L[xδ C(t)]. The root locus analysis method was used to synthesize and tuning the linear control laws. 3.2 Rotational subsystem controller From equations (23-24) it is possible to synthesize a linear controller for stabilizing the rotational subsystem, whose evolution depends directly on input moments τB and τCB. To deal with sustained disturbances motivated by gravity effects and changing thrust input, a feedforward term is supplied to the controller output for the camera’s tilt angle (Eq.26)(Fig. 2). The control design process must provide stability and quick system response to the inner-loop (rotational subsystem) to guarantee the effectiveness of the outer-loop control (translational IFAC ACNAAV 2015 June 10-12, 2015. Seville, Spain 152 J.G.Villagómez et al. / IFAC-PapersOnLine 48-9 (2015) 150–155 153 by defining the generalized coordinates vector qLas qL=[xBzBθBθCB ](9) The equation (eq. 8) is used for computing the multi-body motion equations, by direct application of the Lagrange formalism:   fx fz τB τCB  =d dt ∂LL ∂˙q L−∂LL ∂qL (10) Since we are interested in the evolution of the compounded system w.r.t. the inertial camera frame position, an additional conversion must be applied to the resulting equations. Hence, by defining the final generalized coordinates vector as q=[xCzCθBθC](11) and rewriting the motion equations by introducing θC=θCB +θB(12) finally, undoing the previous translation between frames according to (Eq. 2), motion equations describing the evolution of the camera frame [xCzCθC]onboard the tPVTOL, are obtained. The total mass of the system is denoted as m=mB+mC, and c(·) and s(·) stands for cos(·) and sin(·), respectively. 2.4 Equations of motion Through the Newton-Euler and Lagrange-Euler formulation, equations modeling the overall motion of the tPV- TOL are derived. Set of equations (2) are used in (1) to get the absolute motion of the camera frame given by (3), considering translational and rotational motion of both systems as two interconnected dynamics (Castillo et al. (2005)). Hence, the translational subsystem evolution is described by the following equations:    fx=m¨xC+mB(12(cθB¨ θB−sθB˙ θB 2)+2(sθC¨ θC+cθC˙ θC 2)) fz=m(¨zC+g)−mB12(sθB¨ θB+cθB˙ θB 2)+mB2(cθC¨ θC−sθC˙ θC 2) (13) and the rotational motion set of equations, which is not directly related with ¨xCand ¨zC, as follows:    τB=τCB +a 1¨ θB+a 3(sθCB ¨ θC+cθCB ˙ θC 2) τCB =a 2¨ θC+a 3(sθCB ¨ θB+cθCB ˙ θB 2)−a4cθCB (14) with: a1=ICMB+mBmC m2 12 a2=IC−mBmC m2 2 a3=mBmC m122a4=m C m2 3. LINEAR CONTROL OF THE TPVTOL In order to evaluate more sophisticated control structures, the study of linear control laws was previously proposed by authors in (Villagomez et al. (2014a)). A classical timescale separation is considered between rotational (fastdynamics inner loop) and translational motion (slow dynamics outer-loop). Both dynamics are partially decoupled and controlled by an attitude and position linear controllers. 3.1 Model asumptions and linearization In hover mode the vehicle will operate in regions where θB0o. It will be used a linearized version of the nonlinear model obtained in (Eqs. 13-14), found by means of Taylor approximation in the vicinity of the nominal operating point. Considering the camera positioner angle around a fixed value (θC0o), the following equilibrium conditions are obtained: θeq B=0 τeq B=τeq CB +m Cg12sθeq B=τeq CB θeq C=0 τeq CB =−m Cg2cθeq C=−m Cg2 feq =mg cθeq B (15) given by the small value increment (small angle assumption) of the generalized coordinates: τB(t)=τeq B+τδ B(t)xC(t)=xeq C+xδ C(t)θB(t)=θeq B+θδ B(t) τCB(t)=τeq CB +τδ CB(t)zC(t)=zeq C+zδ C(t)θC(t)=θeq C+θδ C(t) f(t)=feq +fδ(t) (16) whatever xeq Cand zeq Ccoordinates. This assumption leads to the following time-invariant linearized representation of the system, i.e. the translational: ¨ xδ C(t)=feq mθδ B(t)−m B12 m¨ θδ B(t) (17) ¨ zδ C(t)=fδ(t) m−m B2 m¨ θδ C(t) (18) and the rotational subsystem: τδ B (t)−τδ CB (t)=a 1¨ θδ B(t)+a 3sθeq CB ¨ θδ C(t) (19) τδ CB(t)=a 2˙ θδ C(t)+a 3sθeq CB ¨ θδ B(t) −a4cθeq CBfδ(t)+mCg2sθeq CBθδ CB(t) (20) Based on equations (16-20) in the time domain, the application of Laplace transformation yield the following transfer functions, which are considered for the control tasks: xC(s) = 1 ms2(feq −mB12s2)θB(s) (21) zC(s) = 1 ms2f(s) −m B m2θC(s) (22) θB(s) = 1 a1s2(τB(s) −τCB(s)) (23) θC(s) = 1 a2s2(τCB(s)+a 4f(s)) (24) where, e.g., xC(s) = L[xδ C(t)]. The root locus analysis method was used to synthesize and tuning the linear control laws. 3.2 Rotational subsystem controller From equations (23-24) it is possible to synthesize a linear controller for stabilizing the rotational subsystem, whose evolution depends directly on input moments τB and τCB. To deal with sustained disturbances motivated by gravity effects and changing thrust input, a feedforward term is supplied to the controller output for the camera’s tilt angle (Eq.26)(Fig. 2). The control design process must provide stability and quick system response to the inner-loop (rotational subsystem) to guarantee the effectiveness of the outer-loop control (translational IFAC ACNAAV 2015 June 10-12, 2015. Seville, Spain 152 Figure 2. Depiction of the proposed linear control scheme: the translational motion of the tPVTOL is controlled thought the heading’s angle (θ B) and input thrust (f). Camera’s tilt-angular position is pre-compensated to reject thrust input changing effects. motion). Controller expressions for the stabilization of the rotational subsystem: MAV’s pitch angle (θ B) CθB(s) = 34.51s + 690.2 0.001667s + 1 (25) and camera’s tilt position: CθC(s) = 20.23s + 404.5 0.001667s + 1 (26) are provided. 3.3 Translational subsystem controller Translational xCcomponent control expression (Eq. 27) is quite constrained for the evidence of a non-minimum phase zero in (Eq. 21), being underactuated by the θB angle and input thrust on its equilibrium value (Eq. 15). Nevertheless, the height of the camera frame can be directly controlled (Eq. 28) since there is direct action over the variables which zChas dependence on. Output signals of the translational subsystem controller, based on PD controllers, will be used as input references in the rotational subsystem. CxC(s) = 0.1515s + 0.0752 0.002536s2+0.1007s + 1 (27) CzC(s) = 21.68s + 86.72 0.008333s + 1 (28) 4. NON-LINEAR CONTROL OF THE TPVTOL In this section we present the nonlinear control strategies, which are based on a hierarchical scheme considering the time-scale separation between rotational and translational dynamics. We propose to use standard Backstepping and Sliding-mode approaches, to control the internal loop (rotational) and outer-loop dynamics. These control techniques were previously considered and tested on the particular case where references are provided in the MAV frame (Villagomez et al. (2014b)). 4.1 Space state representation The model (Eqs. 13-14) shall be rewritten in a state-space form ˙ x=f(x,u) by introducing x=[x1...x8]∈R8as the state vector of the system: x1=θBx5=xC x2=˙ θBx6=˙xC x3=θCx7=zC x4=˙ θCx8=˙zC (29) It is possible to obtain a reduced model, discarding effects provided by lateral accelerations or centripetal forces in near-hovering flights conditions. These assumptions yield a simplification of the space-state system representation, considered for the control law design, which is stated through the following vector: ˙ x=             x2 1 b1(u3−c(x1−x3)a3x2 2+x2 4−a4) x4 1 a2 (u4+c(x1−x3)a4−a3x2 2) x6 1 m(u1+a 5sx1x2 2−a6cx3x2 4) x8 1 m(u2−gm+mB12cx1x2 2+2sx3x2 4)             (30) where: a5=mB12 a6=mB2b1=a 1a2−a2 3s (x1−x3)2 and the control input vector stated as: u=  u1 u2 u3 u4  =  fx fz τB τCB  =   f·sin(θB) f·cos(θB) (f2−f1) τCB   (31) In order to track a desired trajectory in the space, the computation of virtual control inputs u1and u2provides the required pitch angle and total input thrust to command the attitude controller and the platform rotors, respectively. Hence, the design of nonlinear controllers is based on a cascade control scheme where the attitude controller provides control inputs u3and u4to the tPVTOL in combination with total input thrust fprovided by the position controller (Fig. 3). 4.2 Backstepping controller design Using the Backstepping approach, based on (Bouabdallah and Siegwart (2005)), it is possible to synthesize the virtual control law forcing the system to follow the desired trajectory, since it is possible to design a method to backstep the control input given the obtained model (Eq. 30). Attitude controller For the first step, the tracking error can be considered as: z1x1 r−x1(32) where, e.g., x1 rdefines the desired heading of the tPVTOL (Eq. 29). By considering the Lyapunov theorem of stability, we define a Lyapunov function on z1positive definite V(z1)=1 2z2 1(33) and its time derivative as: ˙ V(z1)=z1(˙x1 r−x2) (34) The stabilization of z1is obtained introducing a virtual control input x2as x2=˙x1 r+α1z1(35) with α1>0. The equation (34) is rewritten as: ˙ V(z1)=−α1z2 1(36) by changing variables: z2=x2−˙x1 r−α1z1(37) For the second step, the following augmented Lyapunov function is considered: V(z1,z 2)=1 2(z2 1+z2 2) (38) The corresponding time derivative of (38) is: IFAC ACNAAV 2015 June 10-12, 2015. Seville, Spain 153 154 J.G.Villagómez et al. / IFAC-PapersOnLine 48-9 (2015) 150–155 ˙ V(z1,z 2)=z2(1 b1(u3−c(x1−x3)a3x2 2+x2 4−a4)) −z2(¨x1 r−α1(z2+α1z1)) −z1(z2−α1z1) (39) Finally, considering zero the reference in the desired acceleration (¨x1 r= 0), the control output u3is then extracted, satisfying ˙ V(z1,z 2)<0: u3=(z1−α1(z2+α1z1)−α2z2)b1+(c(x1−x3)(a3x2 2+a3x2 4−a4)) (40) Term α2z2is added in order to stabilize z1. A similar procedure is performed to compute the input control u4: u4=(z3−α3(z4+α3z3)−α4z4)a 2−(c(x1−x3)(−a3x2 2+a4)) (41) with: z3=x3 r−x3z4=x4−˙x3 r−α3z3(42) Position controller The procedure followed to compute control signals u1and u2is similar to the previous section. u1=(z5−α5(z6+α5z5)−α6z6)m−(a5sx1x2 2)+(a6x2 4cx3) (43) u2=(z7−α7(z8+α7z7)−α8z8)m−12mBcx1x2 2 −2mBsx3x2 4+gm(44) with: z5=x5 r−x5z6=x6−˙x5 r−α5z5 z7=x7 r−x7z8=x8−˙x7 r−α7z7 (45) Similarly, terms α6z6and α8z8with (α6,α 8)>0 are added to stabilize virtual control laws z5and z7, respectively. 4.3 Sliding-mode controller design The first step concerning the design of a Sliding-mode based control law is similar to the one used for the backstepping approach, using the state-space system defined in (Eq. 30). Attitude controller For the rotational subsystem, instead of using a second virtual control variable as z2, the surface s2is used to be consistent with the Sliding-mode law definition: s2x2−˙x1 r−α1z1(46) For the second step, we consider the augmented Lyapunov function: V(z1,s 2)=1 2(z2 1+s2 2) (47) The chosen law for the attractive surface is the time derivative of (46) satisfying (s˙s<0): ˙s2=−k1µ(s2)−k2s2=˙x2−¨x1 r−α1˙z1 =1 b1(u3−c(x1−x3)(a3x2 2+a 3x2 4−a4)) −¨x1 r+α1(s2+α1z1) (48) The control signal u3is then extracted: u3=(−α2 1z1−k1µ(s2)−k2s2)b 1+c(x1−x3)(a3x2 2+a3x2 4−a5) (49) And the same step is followed in order to get u4: u4=(−α2 2z3−k3µ(s4)−k4s4)a 2−c(x1−x3)(−a3x2 2+a4) (50) with: z3=x3 r−x3s3=x4−˙x3 r−α2z3(51) Param. gm Cm BICMBIC1211 Value 9.81 0.4 0.955 0.43 0.25 0.10 0.10 0.03 Unit m s2kg kg ·m2m Table 1. System model parameter values Figure 3. Overview of the proposed control scheme and the decomposition of coupled subsystems. Position controller Following a similar procedure, control signals u1and u2are then computed and extracted: u1=(−α2 3z5−k5µ(s6)−k6s6)a 2−(a5sx1x2 2−a6cx3x2 4) (52) u2=(−α2 4z7−k7µ(s8)−k8s8)a 2 −(mB(12cx1x2 2+2sx3x2 4)−gm) (53) with: z5=x5 r−x5s4=x6−˙x5 r−α3z5 z7=x7 r−x7s5=x8−˙x7 r−α4z7 (54) 5. CONTROL STRATEGIES PERFORMANCE The main aim is to control the camera frame position ([xCzC]=[x5x7]) and its tilt angle (θC=x3). The proposed control strategies will be tested by simulation, considering system parameter values collected in table (Tab. 1), close to real aerial platforms. The evaluation of the linear control strategies is performed following the control scheme depicted in (Fig. 2). Corresponding preliminary results obtained by authors in (Villagomez et al. (2014a)) are to be assessed with the full space-state system representation (13-14) and the nonlinear controllers proposed in (Villagomez et al. (2014b)) following the control scheme depicted in (Fig. 3). Neither disturbance nor uncertainty terms are considered for the simulation process. Propellers forces and control inputs for the translational subsystem fxand fzwould be limited, with a value not exceeding fmax =2mg = 26.58 N. Similarly, the maximum torque provided by the camera positioner rotational joint not exceeds |τCB|≤4mCg2=1.56 N ·m. The heading reference angle computed by the translational controller will be bounded: |θB r|<0.34 rad. Function µ(x) is proposed as an alternative to the non-continuous sign(x) function (Villagomez et al. (2014b)). Nonlinear controller parameters of the Backstepping (Tab. 2) and Sliding-mode (Tab. 3) techniques were simultaneously adjusted by observing the behaviour of the system in closed loop. Initial conditions are [xCzC] 0= [0 0](m) for the translational subsystem, and [θ BθC] 0= [0 0](rad) for the rotational case. The goal for the control task was to achieve the desired position [xC rzC r]fully stabilized in a desired orientation (θC r). 5.1 Attitude controller Rotational subsystem evolution in closed loop (Fig. 4) with input changing references is satisfactory in terms of system IFAC ACNAAV 2015 June 10-12, 2015. Seville, Spain 154 J.G.Villagómez et al. / IFAC-PapersOnLine 48-9 (2015) 150–155 155 ˙ V(z1,z 2)=z2(1 b1(u3−c(x1−x3)a3x2 2+x2 4−a4)) −z2(¨x1 r−α1(z2+α1z1)) −z1(z2−α1z1) (39) Finally, considering zero the reference in the desired acceleration (¨x1 r= 0), the control output u3is then extracted, satisfying ˙ V(z1,z 2)<0: u3=(z1−α1(z2+α1z1)−α2z2)b1+(c(x1−x3)(a3x2 2+a3x2 4−a4)) (40) Term α2z2is added in order to stabilize z1. A similar procedure is performed to compute the input control u4: u4=(z3−α3(z4+α3z3)−α4z4)a 2−(c(x1−x3)(−a3x2 2+a4)) (41) with: z3=x3 r−x3z4=x4−˙x3 r−α3z3(42) Position controller The procedure followed to compute control signals u1and u2is similar to the previous section. u1=(z5−α5(z6+α5z5)−α6z6)m−(a5sx1x2 2)+(a6x2 4cx3) (43) u2=(z7−α7(z8+α7z7)−α8z8)m−12mBcx1x2 2 −2mBsx3x2 4+gm(44) with: z5=x5 r−x5z6=x6−˙x5 r−α5z5 z7=x7 r−x7z8=x8−˙x7 r−α7z7 (45) Similarly, terms α6z6and α8z8with (α6,α 8)>0 are added to stabilize virtual control laws z5and z7, respectively. 4.3 Sliding-mode controller design The first step concerning the design of a Sliding-mode based control law is similar to the one used for the backstepping approach, using the state-space system defined in (Eq. 30). Attitude controller For the rotational subsystem, instead of using a second virtual control variable as z2, the surface s2is used to be consistent with the Sliding-mode law definition: s2x2−˙x1 r−α1z1(46) For the second step, we consider the augmented Lyapunov function: V(z1,s 2)=1 2(z2 1+s2 2) (47) The chosen law for the attractive surface is the time derivative of (46) satisfying (s˙s<0): ˙s2=−k1µ(s2)−k2s2=˙x2−¨x1 r−α1˙z1 =1 b1(u3−c(x1−x3)(a3x2 2+a 3x2 4−a4)) −¨x1 r+α1(s2+α1z1) (48) The control signal u3is then extracted: u3=(−α2 1z1−k1µ(s2)−k2s2)b 1+c(x1−x3)(a3x2 2+a3x2 4−a5) (49) And the same step is followed in order to get u4: u4=(−α2 2z3−k3µ(s4)−k4s4)a 2−c(x1−x3)(−a3x2 2+a4) (50) with: z3=x3 r−x3s3=x4−˙x3 r−α2z3(51) Param. gm Cm BICMBIC1211 Value 9.81 0.4 0.955 0.43 0.25 0.10 0.10 0.03 Unit m s2kg kg ·m2m Table 1. System model parameter values Figure 3. Overview of the proposed control scheme and the decomposition of coupled subsystems. Position controller Following a similar procedure, control signals u1and u2are then computed and extracted: u1=(−α2 3z5−k5µ(s6)−k6s6)a 2−(a5sx1x2 2−a6cx3x2 4) (52) u2=(−α2 4z7−k7µ(s8)−k8s8)a 2 −(mB(12cx1x2 2+2sx3x2 4)−gm) (53) with: z5=x5 r−x5s4=x6−˙x5 r−α3z5 z7=x7 r−x7s5=x8−˙x7 r−α4z7 (54) 5. CONTROL STRATEGIES PERFORMANCE The main aim is to control the camera frame position ([xCzC]=[x5x7]) and its tilt angle (θC=x3). The proposed control strategies will be tested by simulation, considering system parameter values collected in table (Tab. 1), close to real aerial platforms. The evaluation of the linear control strategies is performed following the control scheme depicted in (Fig. 2). Corresponding preliminary results obtained by authors in (Villagomez et al. (2014a)) are to be assessed with the full space-state system representation (13-14) and the nonlinear controllers proposed in (Villagomez et al. (2014b)) following the control scheme depicted in (Fig. 3). Neither disturbance nor uncertainty terms are considered for the simulation process. Propellers forces and control inputs for the translational subsystem fxand fzwould be limited, with a value not exceeding fmax =2mg = 26.58 N. Similarly, the maximum torque provided by the camera positioner rotational joint not exceeds |τCB|≤4mCg2=1.56 N ·m. The heading reference angle computed by the translational controller will be bounded: |θB r|<0.34 rad. Function µ(x) is proposed as an alternative to the non-continuous sign(x) function (Villagomez et al. (2014b)). Nonlinear controller parameters of the Backstepping (Tab. 2) and Sliding-mode (Tab. 3) techniques were simultaneously adjusted by observing the behaviour of the system in closed loop. Initial conditions are [xCzC] 0= [0 0](m) for the translational subsystem, and [θ BθC] 0= [0 0](rad) for the rotational case. The goal for the control task was to achieve the desired position [xC rzC r]fully stabilized in a desired orientation (θC r). 5.1 Attitude controller Rotational subsystem evolution in closed loop (Fig. 4) with input changing references is satisfactory in terms of system IFAC ACNAAV 2015 June 10-12, 2015. Seville, Spain 154 Parameter α1α2α3α4α5α6α7α8 Value 30 15 20 10 20 15 8.5 5 Table 2. Proposed parameter values for the Backstepping controller. Control input signal u1u2u3u4 α35α43α14α25 k55k710 k13.5 k34 k67k810 k23.5 k44 Table 3. Proposed parameter values for the Sliding-mode controller. Figure 4. Simulation: evolution of the θBand θCcomponent in closed loop with linear and nonlinear techniques. time response and zero error in stationary. Numerical results obtained show that the proposed controllers are able to stabilize rotational subsystem. 5.2 Position controller The application of the nonlinear control techniques leads the system to avoid disturbances motivated by the coupling dynamics with more conservative control commands than the linear case. Although the trajectory-tracking objective is fulfilled, the system behaviour in closed loop with nonlinear schemes is more satisfactory than those obtained previously by the authors in (Villagomez et al. (2014a)), as depicted in figure (Fig. 5). 6. FINAL REMARKS AND PERSPECTIVES This paper addressed the planar modeling and control of a multi-body aerial platform consisting of a MAV and a camera positioner. The aim is to use the two additional degrees of freedom provided by a camera positioner in combination with an under-actuated quad-rotor aircraft to get a fully actuated system, allowing free arbitrary movement of the camera in the space. The application of Figure 5. Simulation: evolution of the translational subsystem xCand zCcomponent in closed loop with the proposed controllers the Backstepping approach provides, in general, satisfactory results in stabilizing the full system dynamics. First tuning of the Sliding-mode controller provides good results in stabilizing inner and outer-loop, but it is highly sensitive to the system model parameters. Upcoming working lines will comprise deeper study of parameter assignment and the effects on the stabilization of the tPVTOL and the modeling of the multi-body aerial platform proposed in this work to the space. REFERENCES Altug, E., Ostrowski, J., and Mahony, R. (2002). Control of a quadrotor helicopter using visual feedback. Proceedings ICRA. IEEE International Conference on Robotics and Automation, 1, 72–77. Bouabdallah, S. and Siegwart, R. (2005). Backstepping and sliding-mode techniques applied to an indoor micro quadrotor. Proceedings ICRA. IEEE International Conference on Robotics and Automation. Castillo, P., Lozano, R., and Dzul, A. (2005). Modelling and Control of Mini-Flying Machines. Springer. Kim, S., Choi, S., and Kim, H. (2013). Aerial manipulation using a quadrotor with a two dof robotic arm. In Intelligent Robots and Systems (IROS), 2013 IEEE/RSJ International Conference on, 4990–4995. doi:10.1109/IROS.2013.6697077. Kondak, K., Huber, F., Schwarzbach, M., Laiacker, M., Sommer, D., Bejar, M., and Ollero, A. (2014). Aerial manipulation robot composed of an autonomous helicopter and a 7 degrees of freedom industrial manipulator. In Robotics and Automation (ICRA), 2014 IEEE International Conference on, 2107–2112. doi:10.1109/ICRA.2014. 6907148. Neff, A., Lee, D., Chitrakaran, V., Dawson, D., and Burg, T. (2007). Velocity control for a quad-rotor UAV fly-by-camera interface. SoutheastCon. Proceedings IEEE, 273–278. Orsag, M., Korpela, C., Pekala, M., and Oh, P. (2013). Stability control in aerial manipulation. In American Control Conference (ACC), 2013, 5581–5586. doi:10.1109/ACC.2013.6580711. Villagomez, J., Vargas, M., Ortega, M., and Rubio, F. (2014a). Planar modeling of an actuated camera onboard a MAV. In Proceedings of the 11th Portuguese Conference on Automatic Control, CONTROLO. Villagomez, J., Vargas, M., and Rubio, F. (2014b). Backstepping and sliding-mode techniques applied to an underactuated camera onboard a rotorcraft mav. In 3rd Workshop on Visual Control of Mobile Robots, Vicomor., 1–7. IFAC ACNAAV 2015 June 10-12, 2015. Seville, Spain 155