A prescribed performance referential control for human-like reaching movement of redundant arms

10th IFAC Symposium on Robot Control International Federation of Automatic Control September 5-7, 2012. Dubrovnik, Croatia

A prescribed performance referential control for human-like reaching movement of redundant arms ⋆ Zoe Doulgeri and Paschalis Karalis Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece (e-mail: [email protected]) Abstract: A novel and low-complexity dynamical approach for controlling reaching movements of redundant manipulators is proposed achieving preset performance attributes on the task space error. The iCub platform is used to compare its performance against a multi-referential solution. Simulation results for a 4 degrees of freedom (dof) planar robot are also provided comparing the proposed approach to another velocity based controller. The proposed scheme is shown to exhibit quasi-straight line paths and bell-shaped velocity profiles as well as path and joint configuration consistency in cyclic movements outperforming alternative solutions. 1. INTRODUCTION Studies of human reaching movements from one point to another showed that there is no explicit trajectory planning that moves the system towards the target but there is rather an implicit set of trajectories made possible by a dynamical system. Thus departing from the usual industrial robotic applications where the system tries to match during execution a ’desired’ preplanned trajectory (in line with the planner-controller separation) a number of dynamical approaches have been proposed for reaching motions in which the target acts as an attractor for the arm, Bizzi et al. (1984), Hersch and Billard (2008), Arimoto and Sekimoto (2006). Evidence from human studies reveal quasi straight line paths and bell-shape velocities (Morasso (1981), Flash and Hogan (1984)) but there is conflicting evidence regarding the frame of reference in which the human reaching motion is coordinated. There is evidence supporting the hypothesis that human reaching motions are coordinated in the task space, Morasso (1981), in the joint space, Lacquaniti et al. (1986) and there is even evidence that human planning is not made in a single frame of reference, Desmurget et al. (1998). In the control design of reaching motions by redundant robotic arms the adoption of a specific frame of reference strategy shapes the problems encountered and severely influences the type and the complexity of the proposed solution. A simple task space feedback control scheme was suggested by Arimoto and Sekimoto (2006) and Tahara et al. (2009) that is able to achieve human-like reaching motions given an appropriate damping shaping in the joint space or internal force adjustment in case of both joint and muscle redundancies. Solutions involving a dynamic trajectory generation in joint space as well as in both the task and joint space in a mixed referential approach have also been proposed by ⋆ This research is co-financed by the EU-ESF and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)Research Funding Program ARISTEIA I

978-3-902823-11-3/12/$20.00 © 2012 IFAC

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Sciavicco and Siciliano (1988), Hersch and Billard (2008). The joint space frame of reference generally involves the solution of the ill-posed inverse kinematics problem induced by the joint redundancies that is usually resolved by the introduction of task priorities or various optimization criteria Nakamura et al. (1987), Nakanishi et al. (2005). Not surprisingly some of the proposed approaches avoid this problem by proposing a joint referential trajectory that requires only the calculation of direct kinematics, Sciavicco and Siciliano (1988). A number of other important issues have been considered in some of the proposed algorithms; specifically, joint limit and other constraint avoidance, robustness to kinematic singularities, computational complexity and solution repeatability. However, the resulting movements although they share some of their properties with human reaching movements, are far from fulfilling the ultimate objective of reproducing a dexterous human-like behavior. In this paper we introduce a novel dynamical approach for reaching movements of redundant manipulators which is based on the generation of a joint referential trajectory that guarantees a prescribed performance for the task space error. The design methodology that allows us to impose certain specifications on the performance of measurable signals in uncertain systems has been introduced lately and is called prescribed performance control, Bechlioulis and Rovithakis (2008). According to this methodology the system output is guaranteed to converge to a predefined arbitrarily small residual set with rate no less than a prespecified value and exhibiting maximum overshoot less than a predetermined precentage. The methodology has already been applied for the prescribed performance control of non redundant robots in the cases of regulation and trajectory following for the joint or task position as well as force, Karayiannidis and Doulgeri (2012), Doulgeri and Droukas (2010), Bechlioulis et al. (2010), Bechlioulis et al. (2012). The hypothesis behind the adoption of the prescribed performance methodology 10.3182/20120905-3-HR-2030.00080

IFAC SYROCO 2012 September 5-7, 2012. Dubrovnik, Croatia

in the proposed scheme for controlling reaching motions of redundant manipulators is that a human drives its arm towards a target location using a simple task based feedback in which error convergence performance is imposed at will irrespective of the initial state, the desired position and the distance between them. The iCub platform, N.Tsagarakis et al. (2007), is used to compare its performance against a multi-referential solution proposed in Pattacini et al. (2010). More results are provided by simulating the dynamic model of a simple 4 dof planar manipulator controlled by an inverse dynamics controller using the proposed and another optimization based joint velocity reference. 2. PROBLEM STATEMENT AND PRELIMINARIES Consider a n dof robot manipulating objects in a mdimensional space with m < n. Let q ∈ ℜn be the vector of the generalized joint variables and p ∈ ℜm the vector of the end-effector position in Cartesian coordinates. The direct kinematics relationship specify in a unique straightforward manner the relationship between these two vectors as p = F (q) (1) where F is a continuous nonlinear function that associates to each q a unique p while the inverse mapping in the redundancy case associates to each p a set of q’s usually containing an infinite number of elements. The corresponding relationship between joint velocities and Cartesian velocities is: p˙ = J(q)q˙ (2) where J(q) is the Jacobian of the forward kinematics map (or the manipulator Jacobian). The above relationship can be inverted as follows q˙ = J † (q)p˙ †

T

(3)

T −1

where J = J (JJ ) denotes the Moore-Penrose pseudoinverse of matrix J yielding a minimum norm joint velocity solution in case of a full row rank Jacobian. In many cases this solution is augmented for optimization purposes by a second term (In − J † J)z expressing the null space solution of (2) where In is the identity matrix of dimension n and z is selected according to the purpose. Let pd be the target location and e(t) = [e1 (t)...em (t)]T be defined as the task space error between the target location and the actual location obtained by (1) given measurements of the joint positions: e(t) = p(t) − pd (4) Let the set of arm configurations which correspond to the target location be qT = {q : F (q) = pd }. The reaching movement control objective in redundant robots is to smoothly bring the arm configuration to the target location pd by ensuring the convergence of q(t) to a point in the set qT . In order to get the dynamic error kinematic model we differentiate (4) yielding e(t) ˙ = J(q)q˙ and consider q˙ as a kinematic control input. Hence we can write: e(t) ˙ = J(q)vr (5) Our goal is to design a kinematic controller vr to force the task space error e(t) to zero with prescribed performance. This means that the end-effector position p(t) should be guaranteed to converge to the target location pd with rate 296

less than a prespecified value exhibiting maximum overshoot less than a sufficiently small preassigned constant. Moreover, all joint velocities should converge to zero. We refer to the above task as the Prescribed Performance Referential Control (PPRC) problem for redundant arms. The rest of this section summarizes preliminary knowledge on prescribed performance control (Bechlioulis and Rovithakis (2008)). By prescribed performance guarantees we mean that each component of e(t) evolves within a predefined region that is bounded by a decaying function of time expressed mathematically by the following inequalities. : −Mi ρi (t) < ei (t) < ρi (t) ∀t in case ei (0) ≥ 0 (6) −ρi (t) < ei (t) < Mi ρi (t) ∀t in case ei (0) ≤ 0 (7) where 0 ≤ Mi ≤ 1 and ρi (t) i = 1..., m are bounded smooth, strictly positive decreasing functions satisfying limt→∞ ρi (t) = ρi∞ > 0 called performance functions. As (6) and (7) imply, only one set of the performance bounds is employed and specifically the one associated with the sign of ei (0). The appropriate selection of the performance function ρi (t) i = 1..., m as well as of the constant Mi i = 1..., m imposes performance bounds for the task space error components as clearly illustrated in Fig. 1 for an exponential performance function: ρi (t) = (ρi0 − ρi∞ ) exp (−li t) + ρi∞ (8) where ρi0 , ρi∞ , li are appropriately chosen positive constants. The constant ρi0 = ρi (0), i = 1..., m is selected such that (6) or (7) is satisfied at t = 0 (i.e. ρi0 > ei (0) in case ei (0) ≥ 0 or ρi0 > − ei (0) in case ei (0) ≤ 0). The decreasing rate of ρi (t) which is related to the constant li in this case introduces a lower bound on the required speed of convergence of ei (t). Finally, the maximum overshoot is prescribed less than Mi ρi0 , and in general, may even become zero by setting Mi = 0. The prescribed performance is introduced via the use of a nonlinear transformation, mapping each error component modulated by the performance function to the transformation error space. More specifically, we define ( ) ei (t) εi (t) = Ti i = 1, ..., m (9) ρi (t) where εi (t) is the transformation error and Ti (·) a smooth, strictly increasing function defining a bijective mapping: Ti : (−Mi , 1) → (−∞, ∞) in case ei (0) ≥ 0 (10) Ti : (−1, Mi ) → (−∞, ∞) in case ei (0) ≤ 0 The following candidate transformation function taken from Karayiannidis and Doulgeri (2012) is used in this work and is illustrated in Fig. 2 :    ei (t)  M +  i ρi (t)   ( )  in case ei (0) ≥ 0 ln    ei (t)  Mi 1 − ρi (t) )  (  ei (t) = Ti ( )  ρi (t)   ei (t)  M 1 +  i ρi (t)  ln   in case ei (0) ≤ 0   ei (t)  Mi − ρi (t) (11) Notice that transformation function (11) satisfies the property Ti (0) = 0 which, in addition to (10), is required in order to prove convergence of the task error to zero to the expense of setting a positive although arbitrarily small value for constants Mi , i = 1, ..., m. Furthermore, notice

IFAC SYROCO 2012 September 5-7, 2012. Dubrovnik, Croatia

2 V˙ = −K (∂T ε(t))

e i (0) ≥ 0

Notice that the right hand side of (16) is negative definite owing to ∂T > 0. Hence ε(t) is bounded and it is furthermore converging to zero. It is also possible to recognize that this convergence is exponential by{noticing [ }] that V˙ ≤ −cV where c = 2λmin (K) mini inf t≥0 ∂Ti2 ). This in turn implies the achievement of prescribed performance for e(t) and its further convergence to zero owing to the additional property of the transformation function T (0) = 0. Furthermore, notice that αi (t) is a bounded positive function satisfying 0 < αi (t) ≤ αi (0) < li with limt→+∞ αi (t) = 0 and that the Jacobian pseudoinverse J † (q) in case of singularity avoidance is bounded as it is consisted of elements that are bounded functions of the joint variables. Hence, the kinematic control input (12), i.e. the reference joint velocity vr is converging to zero. Remark 1. Configuration singularities is an important issue in inverse kinematics solutions and can be overcome by various method proposed in the literature. For example, following the damped least square inverse method, Chiaverini et al. (1994), J † (q) = J T (wI + JJ T )−1 with w being a positive weight constant, always provides bounded solutions and can also be used in the proposed method instead of the Moore-Pensrose pseudoinverse in (12).

ρi (t) ρi∞ e i (t) 0

−Mi ρi (t)

0 M i ρi (t) 0 −ρi∞ e i (t)

−ρi (t)

e i (0) ≤ 0 0

Time

Fig. 1. Performance limits ei (0) ≥ 0

ei (0) ≤ 0

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εi (t)

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4. CONTROLLER EVALUATION −Mi

0

0.5 ei (t) ρi (t)

1

−1

−0.5

ei (t) ρi (t)

0

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Mi

Fig. 2. Natural logarithm transformation with T (0) = 0 that since ρi (0), i = 1..., m is selected such that (6) or (7) is satisfied at t = 0, and Mi > 0, i = 1, ..., m, εi (0) i = 1..., m is finite. Owing to the properties of the error transformation, the prescribed performance of each ei (t) in the sense of (6), (7) is satisfied for all t ≥ 0, provided that εi (t) defined through (9) that satisfies (10) is bounded. 3. THE PRESCRIBED PERFORMANCE REFERENTIAL CONTROLLER Let us define ai (t), ∂Ti , i = 1, ..., m as αi (t) , − ρρ˙ii (t) (t) > 0 and ∂Ti ,

∂Ti 1 ∂(ei /ρi ) ρi

> 0 and the transformed error vector T

as ε(t) = [ε1 (t)...εm (t)] . A solution to the considered PPRC problem for redundant arms is provided by selecting the following kinematic control input: vr = −J † (q) [α(t)e(t) + K∂T ε(t)] (12) where ∂T = diag [∂Ti ], α(t) = diag [αi (t)] and K a diagonal matrix of positive control gains. Substituting (12) in (5) we obtain the closed loop error system: e(t) ˙ = − [α(t)e(t) + K∂T ε(t)] (13) Left multiplication of (13) by ∂T using ε(t) ˙ = ∂T [e(t) ˙ + α(t)e(t)] (14) and the commutative property which holds among diagonal matrices yields: ε(t) ˙ = −K∂T 2 ε(t) (15) Consider now the positive definite and radially unbounded function V = 0.5ε(t)T ε(t). The time derivative of V along the trajectories of (15) yields: 297

The iCub platform was used for evaluating the performance of the proposed approach and comparing it with the solution proposed in Pattacini et al. (2010). The latter is an enhanced version of the VITE multi-referential approach yielding a solution resembling a minimum jerk profile both in task and joint space (MinJerk). The iCub is a 53 dof humanoid robot with highly complex arms and hands; there are 7 actuated joints in the arm and 9 in the hand. All the software running on the iCub is open-source and downloadable and in that respect it has been valuable for testing our proposed solution in a real platform. The following test scenarios are presented in this work; (a) a point to point motion of the iCub arm for a long, medium and short distance of 21.7cm, 13.97cm and 10.40cm respectively (b) a return trip between two points (for the medium distance). Accuracy, human-likeness and configuration variability in return trips were parameters of interest. The initial and final points for the three distances are as follows: short distance: p0 = [0.2454 0.1244 0.5359]T and pd = [0.25 0.03 0.58]T , medium distance: p0 = [0.2454 0.1244 0.5359]T , pd = [0.25 0.0 0.6]T , long distance of: p0 = [0.22 0.25 0.54]T , pd = [0.27 0.05 0.60]T . For the PPRC the exponential performance function is used and the following performance parameters: ρi0 = 0.5, ρi∞ = 0.05, li = 0.25, Mi = 1, i = 1, 2, 3 for all distances while control gains are set as follows K = diag[6 6 2]. Figures 3 to 8 show the paths and Cartesian velocity profiles generated by the two methods with a 50ms data report cycle, while Table 1 shows the achieved accuracy and the percentage deviation from a straight line path. Responses of the task space error coordinates for the PPRC are shown in Figure 9 for all distances. Notice how the PPRC outperforms the MinJerk solution in terms of accuracy and the bell-shaped Cartesian velocity response while achieving comparable results to MinJerk regarding

IFAC SYROCO 2012 September 5-7, 2012. Dubrovnik, Croatia

0.2

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Fig. 5. End-effector path in a medium distance; MinJerk (blue), PPRC (turquoise) the deviation form a straight line path. The task space error respects performance bounds as shown in Figure 9. Results from a point to point return trip for the medium distance using both PPRC and MinJerk are shown in Figures 10 to 13. It is clear that the proposed solution is capable of producing comparable paths and Cartesian velocity profiles in the forward and backward movement but more importantly return close to the initial joint configuration while the corresponding MinJerk responses are characterized by large deviations between the forward and backward movements. Hence. the proposed scheme is 298

0 0

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Fig. 8. Cartesian velocity profile in short distance; MinJerk (blue), PPRC (turquoise) shown to be superior in terms of accuracy, human-likeness and configuration variability. Simulation results from a 4 dof planar robot dynamic model with robot data taken from Arimoto and Sekimoto (2006) are also provided. The initial and target locations of the robot end-effector are p0 = [−0.0381 0.5295] and pd = [−0.3 0.4]. A third degree polynomial performance function, inspired by the min jerk trajectory formulation, is utilized: ρi (t) = ki0 + ki1 t + ki2 t2 + ki3 t3 for t ≤ T where

IFAC SYROCO 2012 September 5-7, 2012. Dubrovnik, Croatia

PPRC MinJerk

Table 1.

60

60

40

40

angle position [deg]

PPRC MinJerk

angle position [deg]

Accuracy measured in cm long medium short 0.046 0.4 0.6 1.15 1.2 1.2 % Deviation from the linear path 11 9 9 9 9 8

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Fig. 11. Joint trajectories with PPRC (a) forward (b) return

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Fig. 12. MinJerk (a)End-effector path and (b)Cartesian velocity profile during a return trip: forward(green), return(purple)

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Fig. 9. Task space error coordinate responses: long distance (blue) medium (purple) short (green), performance boundary (red)

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Fig. 13. Joint trajectories with MinJerk (a) forward (b) return

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Fig. 10. PPRC (a)End-effector path and (b)Cartesian velocity profile during a return trip: forward(green), return(purple) ki0 = ρi0 , ki1 = 0, ki2 = −3(ρi0 − ρi∞ )/T 2 , ki3 = 2(ρi0 − ρi∞ )/T 3 and ρi (t) = ρi (T ) = ρi∞ for t > T for a choice of T > 0. The following parameter values were used in the simulation ρi0 = 0.35, ρi∞ = 0.05 for i = 1, 2 and T = 1.4. i0 −ρi∞ ) Notice that ρi (t) > 0 and 0 ≤ ai (t) ≤ 3(ρ2T as ρi∞ required by the prescribed performance control approach. The control gain K in (12) is: K = 10−3 I2 where I2 is the identity matrix of dimension 2. Comparisons are made with a velocity based control scheme (VBC) with posture 299

optimization from some preferred posture qrest (Nakanishi et al. (2005)). Specifically vr = J † p˙r −α(I4 −J † J)∇g where g(q) = 12 (q − qrest )T (q − qrest ) and p˙r = p˙d + Kp (pd − p). Both controllers are utilized within the inverse dynamics control law: τ = M (q)v˙ r +C(q, q)+G(q)+K ˙ ˙ Gain v (vr − q). values are Kv = 0.5I4 and Kp = 20I2 and qrest is chosen to be equal to the initial posture, qrest = [59 43 25 95] deg. As shown in Figures 14 to 16 the proposed controller prevails in all aspects of performance and human-likeness. 5. CONCLUSIONS The velocity controller proposed in this work guarantees prescribed performance of the task space error in reaching movements of redundant manipulators. Evaluation results with the iCub platform show its superiority with respect to a MinJerk multi-referential solution in terms of accuracy and bell-shaped Cartesian velocity responses, while producing quasi-straight line paths. It is further capable of producing similar paths, Cartesian velocity profiles and postures in return trips, features that are not shared by

IFAC SYROCO 2012 September 5-7, 2012. Dubrovnik, Croatia

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Fig. 14. Simulation results for a 4-dof planar robot (a)End-effector path (b) Cartesian velocity profile ; PPRC(blue), VBC(green) 300 anglular velocity [deg/sec]

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error[m]

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Fig. 16. Task space error response. x coordinate(blue), y coordinate(green); PPRC(solid line), VBC (dashed line); performance bound(red dashed line) the MinJerk solution. Simulations of a 4-dof planar robot using the proposed scheme and a velocity based scheme with posture optimization via an inverse dynamic control law show its superiority in terms of human-likeness. Future effort is devoted in the design of a model free control law that stabilizes the closed-loop system dynamics. REFERENCES Arimoto, S. and Sekimoto, M. (2006). Human-like movements of robotic arms with redundant dofs:virtual spring-damper hypothesis to tackle the bernstein problem. In Proc. IEEE International Conference in Robotics and Automation, 1860–1866. Bechlioulis, C., Doulgeri, Z., and Rovithakis, G. (2010). Neuro-adaptive force/position control with prescribed performance and guaranteed contact maintenance. Neural Networks, IEEE Transactions on, 21(12), 1857 – 1868. Bechlioulis, C., Doulgeri, Z., and Rovithakis, G. (2012). Guaranteeing prescribed performance and contact main300

tenance via an approximation free robot force/position controller. Automatica, 48(2), 360 – 365. Bechlioulis, C. and Rovithakis, G. (2008). Robust adaptive control of feedback linearizable mimo nonlinear systems with prescribed performance. IEEE Transactions on Automatic Control, 53(9), 2090–2099. Bizzi, E., Accornero, N., Chapple, W., and Hogan, N. (1984). Posture control and trajectory formation during arm movement. The Journal of Neuroscience, 4, 2738– 2744. Chiaverini, S., Siciliano, B., and Egeland, O. (1994). Review of the damped least-squares inverse kinematics with experiments on an industrial robot manipulator. IEEE Transactions on Control System Technology, 2(2), 1–10. Desmurget, M., Pelisson, D., Rosseti, Y., and Prablanc, C. (1998). From eye to hand: Planning goal-directed movements. Neuroscience and Biobehavioural Reviews, 22(6), 761–788. Doulgeri, Z. and Droukas, L. (2010). Robot task space pid type regulation with prescribed performance guaranties. In Proc. IEEE/RSJ International Conference onIntelligent Robots and Systems, 1644–1649. Flash, T. and Hogan, N. (1984). The coordination of arm movements: An experimentally confirmed mathematical model. Journal of Neuroscience, 5, 1688–1703. Hersch, M. and Billard, A. (2008). Reaching with multireferential dynamical systems. Auton. Robot, 25, 71–83. Karayiannidis, Y. and Doulgeri, Z. (2012). Model-free robot joint position regulation and tracking with prescribed performance guarantees. Robotics and Autonomous Systems, 60, 214–226. Lacquaniti, F., Soechting, J., and Terzuolo, S. (1986). Path constraints on point to point arm movements in three dimensional space. Neuroscience, 17(2), 313–324. Morasso, P. (1981). Spatial control of arm movements. Experimental Brain Research, 42(2), 223–227. Nakamura, Y., Hanafusa, H., and Yoshikawa, T. (1987). Task-priority based redundancy control of robot manipulators. The International Journal of Robotics Research, 6(2), 3–15. Nakanishi, J., Mistry, M., and Schaal, S. (2005). Comparative experiments on task space control with redundancy resolution. In in IEEE International Conference on Intelligent Robots and Systems, 1575–1582. IEEE. N.Tsagarakis, G.Metta, Sandini, G., and et al, D.V. (2007). icub: the design and realization of an open humanoid platform for cognitive and neuroscience research. Advanced Robotics, 21(10), 1150–1175. Pattacini, U., Nori, F., Natale, L., Metta, G., and Sandini, G. (2010). An experimental evaluation of a novel minimum-jerk cartesian controller for humanoid robots. In Proc. IEEE/RSJ International Conference onIntelligent Robots and Systems, 1668–1674. Sciavicco, L. and Siciliano, B. (1988). A solution algorithm to the inverse kinematic problem fpr redundant manipulatotrs. IEEE Journal of Robotics and Automation, 4(4), 403–410. Tahara, K., Arimoto, S., Sekimoto, M., and Luo, Z. (2009). On control of reaching movements for musculoskeletal redundant arm model. Applied Bionics and Biomechanics, 6(1), 57–72.