Real Time Direct Kinematic Problem Computation of the 3PRS robot Using Neural Networks

Real Time Direct Kinematic Problem Computation of the 3PRS robot Using Neural Networks Asier Zubizarreta, Mikel Larrea, Eloy Irigoyen, Itziar Cabanes, Eva Portillo Automatic System and Control Engineering Dpt., University of the Basque Country (UPV/EHU) Abstract Getting a reliable calculation of the Direct Kinematic Problem (DKP) is one of the main challenges for the implementation of Real-Time (RT) controllers in Parallel Robots. In the general case, the solution of the DKP needs, among other calculated variables, the estimation of the robot pose in terms of the sensors placed on actuators. Therefore, obtaining all these variables requires the use of iterative procedures which employ high computational time. Artificial Neural Networks have been proposed to implement the complex DKP equation mapping in the literature due to their universal approximator property. However, the proposals in this area do not consider the Real Time implementation of the ANN based solution, and no approximation error vs computational time analysis is carried out. In this work, a methodology that uses Artificial Neural Networks (ANNs) to approximate the DKP is proposed. Based on the 3PRS parallel robot, a comprehensive study is carried out in which several network configurations are proposed to approximate the DKP. Moreover, to demonstrate the effectiveness of the approach, the proposed networks are evaluated considering not only their approximation capabilities, but also their Real Time performance in comparison with the traditional iterative procedures used in robotics. Keywords: Parallel Robots, Kinematic Problem, Artificial Neural Network 1. Introduction Current robotic applications demand more precision, speed and load handling capabilities in order to fulfil productivity and economic goals. The classical mechanical structure used for these applications is the single kinematic chain approach of serial robots, which is composed by a single series of elements that connect a fixed base with the end effector or tool. This structure provides wide operational workspace and flexibility, although its load handling capability and dynamic performance are usually limited. IThis work was supported in part by the Government of Spain under project DPI2012-32882 and UPV/EHU under grant UFI11/28 Email addresses: [email protected] (Asier Zubizarreta), [email protected] (Mikel Larrea), [email protected] (Eloy Irigoyen), [email protected] (Itziar Cabanes), [email protected] (Eva Portillo) Preprint submitted to Neurocomputing May 2, 2016 This is the Manuscript version of a Published Work that appeared in final form in Neurocomputing 271: 104-114 (2018) To access the final edited and published work see https://doi.org/10.1016/j.neucom.2017.02.098 © 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/ In order to satisfy industry’s requirements, in the last decade, the interest on alternative robotic structures has grown. Parallel Robots [1] have been proposed as a suitable approach for high dynamic performance tasks, i.e., high speed and acceleration, precision or high load handling tasks. Their outstanding performance in these areas is due to their multiple kinematic chains, or parallel structure, in which a moving platform, where the tool is located, is connected to a fixed one by means of several serial links. This structure provides higher stiffness than the one of serial robots, allowing load to be distributed among the limbs. Moreover, positioning errors can be compensated within the multiple limbs, and the structure can even be designed to present lower inertia by locating the actuators in the fixed base. However, the complex, multiple loop structure of parallel robots presents some disadvantages. For instance, their workspace is much smaller than the one of serial robots, and their structure is due to present singularities in this workspace. One of the most important handicaps is related to the control of these robots, as the Direct Kinematic Problem (DKP) needs to be solved. The DKP calculates the pose of the Tool Centre Point (TCP) of the robot in terms of the motion of the actuators, and is mandatory for proper robot control. In the general case, however, this problem has no analytical solution due to the complexity of the equations, and iterative numerical procedures have to be used to estimate the position of the end effector in each control loop. Newton-Raphson (N-R) approach is widely used for the calculation of the DKP [2]. This procedure requires an initial guess of the end effector location and an indefinite number of iterations, converging to a local solution that satisfies the complex kinematic equations of the robot. The number of iterations is highly dependant on the initial guess and its distance to the solution, and for each iteration, the Jacobian of the equation system has to be calculated and inverted. Hence, the computational cost of this approach is usually high, which represents a key issue when implementing advanced controllers for parallel robots. Being high speed tasks one of the main areas of application of parallel robots, several works have focused on optimising the computational cost of the DKP. Three main groups can be detected among the proposed solutions. The first group of approaches is focused on reducing the complexity of the nonlinear kinematic model of parallel robots by obtaining an univariate polynomial that can be solved efficiently. Examples of application can be found for several parallel robots, such as the 4 degrees-of-freedom (dof) Sch¨onflies platform detailed in [3], or the 3 dof reconfigurable MaPaMan [4], being the most studied one the 6 dof Gough-Stewart platform [5, 6, 7]. Although effective, these approaches are very sensitive to measurement errors in practice, as a small set of data is used to estimate the rest of the variables. The second group is focused on introducing extra sensor data that allows an analytical solution of the DKP. For that purpose, additional sensors have to be introduced in the mechanical structure in order to measure non-actuated joint motion. This approach has demonstrated to reduce DKP computational cost significantly while maintaining high estimation accuracy [8, 9]. However, the use of extra sensors increases the cost of the robot, and adds additional complexity to its calibration. Finally, the third group is focused on the use of approximators that simplify the highly coupled and nonlinear kinematic equations. The main hypothesis behind these approaches 2 is that a bounded but not exact accuracy is needed when estimating the end effector pose. Hence, universal approximators can be used if the approximation error is kept within the required bounds. In this context, the use of Taylor series was proposed by Wang, et al [10] to calculate the DKP of the Gough-Stewart platform. Among other approaches, the use of Artificial Neural Networks (ANN) have also been proposed due to their nature as universal approximators [11]. ANNs simulate the behaviour of biological neural networks by means of a set of mathematical representations of neurons, which are grouped in interconnected layers. Each neuron combines its multiple inputs to compute its output by means of a nonlinear activation function, which allows them to learn high complex input-output data relations such as the kinematic relations of parallel robots. Few works have focused on the use of ANN as approximators for the DKP solving in parallel robots. In these, architectures based on Radial Basis Function (RBF) [12, 13], polynomial neural networks [14] or Adaptive-network-based fuzzy inference system [15] have been proposed. However, the most popular one is the Multi Layer Perceptron (MLP) architecture, with applications in the calculation of the DKP of several parallel robots, such as the Hexa [16], the Hexapod [17, 18], the 3RRR planar robot [13] or the 3RRR spherical robot [19]. The aforementioned works demonstrate the potential of ANNs as an effective solution to the computation of the DKP in parallel robots. However, most of the works are focused on the quality of the estimation, and do not consider two important issues: the definition of the bounded error to determine the validity of the ANN configuration, and the Real Time computational cost of the ANN based approach in comparison with traditional ones [20]. This work is focused on the Real Time implementation of ANN to solve the DKP of parallel robots, providing two main contributions in this area: a) a methodology to determine the best ANN architecture is provided based on a bounded error criteria; and b) a Real Time computational cost study is carried out to determine the effectiveness of the approach and to discuss its validity for RT control applications. For that purpose, the 3PRS parallel robot has been selected as the study case and is presented in Section 2. The DKP solving approaches based on ANN are discussed in Section 3. The ANN training procedure and approximation performance is analysed in Section 4, in which the selection criteria based on bounded accuracy error is discussed. The Real Time performance of the proposed networks is evaluated in Section 5. Finally, the most important ideas are summarised. 2. The 3PRS Parallel Robot The 3PRS is a lower mobility parallel robot, composed by three PRS (prismatic-rotaryspherical) limbs that connect a fixed base with a moving platform where the end effector, or tool, is attached (Fig. 1). This structure provides three degrees of freedom: displacement along the global zaxis, and two independent rotary motions θxand θyalong the global x and yaxes. However, due to the configuration of the robot, movement in the xand yaxes and the θzangle is not null. This effect is known as parasitic motion [21]. 3 hp hp ra Figure 1: 3PRS Parallel Robot There exist several applications for this robot, such as dish positioning, solar tracking for energy generation, machining or testing devices. In order to achieve the desired mobility, the robot is actuated by three linear motors AiBithat are attached to the fixed base in a symmetrical configuration. Three limbs BiCitransfer the motion of the actuators to the moving platform where the TCP Pis located. For that purpose, one end of the limb, Bi is composed by a slider and a rotary joint, while the other, Ci, presents a spherical joint. This configuration restricts the motion of the limb to the πiplane (Fig. 2), defined by Ai, Biand Ci. The moving platform has a regular triangular geometry, and its height is hp. The kinematic relations of the 3PRS parallel robot are based on the loop closure equations (Fig. 2), which relate the motion of the actuators qaand the pose of the end effector located in the moving platform. p+dO i−li−bi−ai=03×1,i= 1,2,3(1) where the first two terms define the pose of the end effector, being p=x y z Tits position in the fixed frame O(x, y, z) and dO ithe orientation vector of each vertex of the moving platform, which is calculated as, dO i=R di(2) which represents the projection of the constant vector di=PCi(defined in the moving frame) in the fixed frame, di=r−r sin (2π(i−1)/3) −hpTi=1,2,3 (3) didefines the geometry of the moving platform, being r= 0.3638 (m) the radius of the moving platform and hp= 0.04 (m) its height. The rotation matrix Rrelates the orientation of the moving frame P(u, v, w) with respect to the fixed one O(x, y, z). If Roll-Pitch-Yaw (θx, θy, θz) notation is selected, 4 R=  uxvxwx uyvywy uzvzwz  =  cθzcθycθzsθysθx−sθzcθxsθzsθx+cθzsθycθx sθzcθycθzcθx+sθzsθysθxsθzsθycθx−cθzsθx −sθycθysθxcθycθx  (4) where cand sstand for cosine and sine trigonometrical operations, respectively. hp hp ra Figure 2: Kinematic loop The three last terms of Eq. 1 are defined in terms of joint variables. liis the vector related to limb BiCi, whose length li= 0.9805 (m) is constant, li=li  −cos (2π(i−1)/3) sin ϕi −sin (2π(i−1)/3) sin ϕi cos ϕi  i= 1,2,3 (5) biis the position vector of the slider joint Bi, which depends on the actuated variables qai=|AiBi|and the constant linear guide angle α=−0.7854 (rad). bi=  qaicos αcos(2π(i−1)/3) qaicos αsin(2π(i−1)/3) qaisin α  i= 1,2,3 (6) Finally, aiis the constant position vector of each of the three linear guides with respect to the fixed frame, defining the fixed platform geometry, ai=airacos (2π(i−1)/3) −rasin (2π, (i−1)/3) 0 Ti=1,2,3 (7) where ra= 0.425 (m) is the radius of the linear guides disposition. If a distance constraint l2 i=||li|| is imposed to each of the vectorial closure loop equations (Eq. 1), the relationship between the output coordinates x=x y z θxθyθzTand 5 the input joint variables associated to the motion of the actuators qa=qa1qa2qa3T can be defined as follows, Γi+3 =q2 ai+Biqai+Ci= 0 i= 1,2,3(8) where Bi= (−2CAix cos α)/cos(2π(i−1)/3) −2CAiz sin α−2CAiy cos αsin(2π(i−1)/3) Ci=||CAi|| +h2 b−l2 i+ (2 hbCAix sin α)/cos(2π(i−1)/3) +2 hbCAiy sin αsin(2π(i−1)/3) −2hbCAiz cos α being CAi=px+R di−ai=CAix CAiy CAiz T, for i= 1,2,3. As stated previously, the 3PRS presents only 3 degrees of freedom, being z,θxand θy the independent output variables. In order to define the kinematic equation system of the robot, the relationship between these independent output variables and the parasite motions x,yand θzneeds to be modelled. Hence, three constraint equations have to be introduced. As the motion of limbs BiCiis constrained to a plane πi(Fig. 2)[21], Γ1=hpwy−r uy−y= 0 Γ2=hpwx+r 2(ux−vy)−x= 0 Γ3=vx−uy= 0 (9) where wx,wy,vx,vy,uxand uydepend on the RPY Euler angles (θx, θy, θz) (Eq.4). The equation system obtained by combining Eqs. 8 and 9 defines the kinematic equations used to solve the Direct Kinematic Problem. 3. Solving the Direct Kinematic Problem As stated previously, in robotic applications the location of the end-effector associated to the moving frame P(u, v, w) with respect to the fixed frame O(x, y, z) is essential to execute robotic tasks. As this measurement is not possible in the general case, the Direct Kinematic Problem needs to be solved, so that the output variables that define the end-effector pose x, can be estimated in terms of the measurable actuator variables qa. Hence, the DKP is, in fact, a mapping problem, x=f(qa) (10) where as detailed in the previous section, xis a vector containing the cartesian position and orientation of the platform and qais the input joint variable vector associated to the motion actuators. However, being highly nonlinear, an analytical expression for the mapping cannot be found in the general case. The classical approach is to solve the mapping locally, by using an iterative solving procedure based on an initial guess, i.e, Newton-Raphson approach. However, it is also possible to try to define an ANN to learn the mapping between the inputs and outputs. These alternatives will be discussed next. 6 3.1. Classical approach: Newton-Raphson Newton-Raphson approach is based on the use of the Jacobian Jto iterate from an initial guess to the solution. In order to apply it to the DKP solving case, the values of the actuated joint variables qaare measured and considered known, and an initial guess of all the variables in x(independent and parasite) is required. Then, the numerical iterative process is defined as follows, xk+1=xk−J−1Γ(xk,qa) (11) where Γ(xk,qa) = 0, for the particular case of the 3PRS robot, is the 6 equation system defined in Eqs. 8 and 9, and J=∂Γ/∂x is the 6 ×6 Jacobian of the parallel robot. The iterative procedure stops when a maximum number of iterations has been reached, or the relative error between iterations is less than a predefined value. Although it quickly converges to the solution, this approach requires inverting the Jacobian matrix of the robot, which can be time consuming. 3.2. Artificial Neural Network approach ANNs are mathematical constructs that emulate the biological functions of neural networks. Their structure is based on artificial neurons, that emulate biological ones by the use of nonlinear activation functions. ANNs have demonstrated to be universal approximators [11], being able to obtain a bounded error in function approximation depending on the size of the hidden layer. This way, they have been used in a wide range of applications. As detailed in the introduction, in the robotics literature the Multi Layer Perceptron (MLP) structure [11] is one of the most used due to the extensive literature existing on its training and performance tuning. Hence, in this work, this architecture is selected to perform the comparative study. It should be noted that ANNs need to be trained so that they learn the mapping problem. This procedure requires to define a set of examples, the Training Set, so that the weights and biases that define the parameters of the ANN can be tuned to fit the mapping problem using the Backpropagation Method. The training of an ANN is not a trivial task, and it usually is the most complex and time consuming one. The particular training procedure of the ANNs for the study case of the 3PRS parallel robot is analysed in Section 4. Once trained, the ANN can reproduce the mapping problem after defining its inputs. Although approximation errors can arise, depending on the quality of the training, the execution of the ANN is relatively fast. The propagation stage in the neural network takes the inputs of the network and multiplies them by the weights connecting to the next layer. Once the neurons of the hidden layer have their weighted inputs added, the result passes through the activation function to propagate the signal to the next layer. All in all, the propagation stage is a set of matrix multiplications, sums and nonlinear activation functions. The computational cost of this procedure depends on the number of neurons and signals to process, and will be analysed in Section 5. For the particular case of the 3PRS parallel robot, and the Direct Kinematic Problem mapping, different ANN topologies can be chosen. An important issue to be considered is 7 the output configuration of the network, i.e., the number of outputs to be estimated. It seems reasonable that a complex mapping problem will require less hidden layer neurons if its outputs are estimated separately by different ANNs. Hence, based on this reasoning, three different ANN based DKP estimator approaches can be defined based on the nature of the three outputs to be obtained (z,θxθy): •Single 3-N-3 structure, that handles the entire mapping problem in its globality (Fig.3), i.e, from the three actuator sensor positions qa1,qa3,qa2, the net estimates the independent output variables (z,θxθy). •Two structures, one 3-N-1 for estimating the output position zand other 3-N-2 that provides the estimation of the output orientation (θx, θy) (Fig.4). •Three 3-N-1 structures, one for each output (Fig.5). The last variable to be determined is the size of the hidden layer (N). However, the lack of an established method to obtain Nrelated to a concrete problem, requires the use of an experimental approach to determine this variable. The procedure to determine the appropriate hidden layer neuron number is analysed in the next section. N input output hidden layer qa1 2 qa 3 qa z θx y θ Figure 3: Single 3-N-3 net 4. ANN training methodology for DKP implementation In order to determine the best alternative among the proposed network configurations (all coordinates 3 −N−3, orientation coordinates in 3 −N−2, and single coordinates 3−N−1), an iterative batch training procedure has been designed. For this reason, for each network configuration a set of neural networks has been evaluated, whose hidden layer neuron number Nvaries from 5 to 90 with intervals of 5 neurons. The initialisation of the net parameters (weights and biases) has been carried out applying Nguyen-Widrow approach. Furthermore, to reduce the effect of random weight initialisation, for each value of N, 10 nets have been trained. According to the universal approximator property as stated in [11], any type of activation function can be implemented in the designed MLP. However, due to the existence of both 8 input output hidden layer N qa1 2 qa 3 qa N θx y θ z Figure 4: Position and Orientation: 3-N-1 z& 3-N-2 (θx,θy) nets N input output hidden layer N N qa1 2 qa 3 qa z θx y θ Figure 5: Three independent 3-N-1 (z,θx,θy) nets positive and negative input-output data, the hyperbolic tangent and linear activation functions have been selected for the hidden and output layer respectively [22] for all networks 9 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 Neurons in hidden layer 0 50 100 150 time (µs) 3-N-3 ANN Time Performance tnorm th to tdnorm 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 Neurons in hidden layer 0 50 100 150 time (µs) 3-N-2 ANN Time Performance tnorm th to tdnorm 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 Neurons in hidden layer 0 50 100 150 time (µs) 3-N-1 ANN Time Performance tnorm th to tdnorm Figure 12: ANN Time Performance vs hidden neuron number Approach Single Position+Orientation Independent N-R Configuration 3-55-3 3-20-1 z3-15-2 3-20-1 z3-15-1 θx3-10-1 θy Time (µs) 80.19 45.24 40.65 45.24 39.77 36.18 675.71 85.89 121.19 Table 3: Total Performance Times for each DKP solving approach 6. Conclusions An efficient calculation of the Direct Kinematic Problem in Parallel Robots is critical for designing Real Time controllers. This effectiveness is related to obtaining precise robot pose values below a maximum error limit, at each sample time of execution. Although the classical approach to solve the DKP is the Newton-Raphson algorithm, paradigms such as ANN have also been proposed. ANNs provide advantages over the NR approach, as their computational cost is lower. However, they provide an approximation to 16 the solution, and a proper training procedure has to be defined to achieve an appropriate accuracy. In this work, a comprehensive study of Multilayer Perceptron Artificial Neural Network accuracy and time performance in DKP solving for a 3PRS parallel robot has been presented. Taking into account the mapping problem presented by this robot, several ANN configurations have been proposed: a single 3-N-3 network, two networks (position and orientation), and a single network for each output. Each configuration has been tested with a number of hidden layer neurons in the range [5-90]. For the training process, a set of examples for ANN tunning (192.238) and ANN testing (2.836.219) have been picked from the 3PRS effective workspace, being different examples in both sets. Furthermore, in order to avoid the dependency of ANN parameters initialisation, each of those structures were trained 10 times, selecting the best iteration as representative. The maximum approximation error has been considered as the performance index in order to define the best networks. To extract conclusions about the goodness of ANN in this problem, the networks have been evaluated in accuracy and in Real-Time, comparing the results with the classical NR method. This way, the best structures for each configuration have been: 3-20-1 for z position, 3-15-1 for θxorientation, and 3-10-1 for θyorientation; 3-15-2 for orientation and 3-20-1 zfor position; 3-55-3 for position and orientation coordinates. It is necessary to note that the worst pose estimation results have been obtained close to the workspace limits, where points with singularities emerge. For further works, a previous study could enhance the distribution of pose coordinates chosen as examples for the training process. Other relevant aspect of this work is the analysis made in computational cost, during execution in a Real time platform. In order to evaluate the performance and efficiency of both methods, N-R and ANN, a deep analysis in time consuming has been made in a Industrial PC. Considering that the introduced code in such platform will consume different amounts of time in each phase, four steps have been evaluated separately. 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