Human-like movement is fundamental for natural human-robot interaction and collaboration. We have developed
in a model for generating arm and hand movements an anthropomorphic robot. This model was inspired by the Posture-Based
Motion-Planning Model of human reaching and grasping movements. In this paper we present some changes to the model we
have proposed in [4] and test and compare different nonlinear constrained optimization techniques for solving the large-scale
nonlinear constrained optimization problem that rises from the discretization of our time-continuous model. Furthermore, we
test different time discretization steps.Eliana Costa e Silva was supported by FCT (grant: SFRH/BD/23821/2005)

Bicho, E.

Costa, M. Fernanda P.

Erlhagen, Wolfram

Silva, Eliana Oliveira Costa

English

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Human-Like Movement of an Anthropomorphic Robot: Problem Revisited

Universidade do Minho: RepositoriUM

Human-Like Movement of an Anthropomorphic Robot:Problem RevisitedE. Costa e Silva1∗, M. F. Costa†, E. Bicho∗ and W. Erlhagen†∗Dept. of Industrial Electronics, University of Minho, Portugal†Dept. of Mathematics and Applications, University of Minho, PortugalAbstract. Human-like movement is fundamental for natural human-robot interaction and collaboration. We have developedin a model for generating arm and hand movements an anthropomorphic robot. This model was inspired by the Posture-BasedMotion-Planning Model of human reaching and grasping movements. In this paper we present some changes to the model wehave proposed in [4] and test and compare different nonlinear constrained optimization techniques for solving the large-scalenonlinear constrained optimization problem that rises from the discretization of our time-continuous model. Furthermore, wetest different time discretization steps.Keywords: Large-scale nonlinear optimization, human-like collision-free arm movementsPACS: 02.60.PnINTRODUCTIONIn previous work we have presented a model for generating smooth and human-like arm and hand movements ofan anthropomorphic robot, ARoS, that is inspired by the Posture-Based Motion-Planning Model (PBMP) of humanreaching and grasping movements [1]. Several experiments have been made on human-robot interaction tasks (see[2, 3]). However, for a ﬂuent and efﬁcient human-robot interaction, movements must be generated in real-time or atleast in an sufﬁciently small interval of time. In [4] we tested different three solvers for determining the solution of thelarge-scale nonlinear constrained optimization problems that rise from the discretization of our time-continuous model.Here we present some changes to the model presented in [4] and use other optimization techniques. Furthermore, weexplore how the discretization steps affects the performance of the solvers both in terms of computational time andquality of the solution.There is a large amount of available optimization software (see e.g. http://www.neos-server.org/),some of which are open-source (see e.g. https://projects.coin-or.org/). Roboticists have not playedtoo much attention to it. An exception is [5], who use IPOPT [6] merely to solve the inverse kinematics problem ofan anthropomorphic robotics arm in point-to-point movements in the absence of obstacles. In contrast, we modeledthe entire trajectory of the arm and hand, which includes obstacle avoidance, in reaching and grasping movements.Additionally, several grip types are consider, speciﬁcally, side and above grips.THE MODELThe model we have developed has been described in [4]. Here we give a brief description and present some changes.A robotic arm (and hand) can be represented as a series of links connected by joints. The number of joints whichcan be independently actuated deﬁne its degrees of freedom (DOFs). ARoS’ anthropomorphic robotic arm has 7 DOFsand its hand has 4 DOFs. The arm and hand conﬁguration in joint space is deﬁned by the vector θ = (θ1,θ2, . . . ,θ11).Taking inspiration from the PBMP by [1], we deﬁne the movement of each joint on the robotic arm and hand as thesuperimposition of two movements: (i) a direct movement, describing a bell-shaped unimodal velocity proﬁle, fromthe initial to ﬁnal posture; (ii) a back-and-forth movement from initial to a bounce posture, intended to avoid collisionwith obstacles in the robot’s workspace. Therefore, the movement planning proposed here for the anthropomorphicrobotic arm and hand can be summarised as the resolution of two subproblems: (Pa) determining the ﬁnal posture, i.e.,1 Eliana Costa e Silva was supported by FCT (grant: SFRH/BD/23821/2005).Numerical Analysis and Applied Mathematics ICNAAM 2011AIP Conf. Proc. 1389, 779-782 (2011); doi: 10.1063/1.3636848©   2011 American Institute of Physics 978-0-7354-0956-9/$30.00779a vector of arm and hand joint angles, θ f ∈R11, that allows ARoS to grasp a given object subject to speciﬁc constraints(e.g. grip type); (Pb) determining a bounce posture, θ b ∈R11, that serves as a sub-goal for a back-and-forth movement.The sequence of joint angles of the robotics arm and hand is given byθ(t,θ f ,θ b) = θ 0 +(θ f −θ 0)(10τ3−15τ4 +6τ5)+ v0T(τ−6τ3 +8τ4−3τ5)+12a0T 2(τ2−3τ3 +3τ4− τ5)+(θ b−θ 0)sin2(π τϑ ), (1)where θ 0,v0,a0 ∈ IR11 are constant vectors representing initial joint position, velocity and acceleration, respectively,T ∈ IR+ represents the movement duration, t ∈ [0,T ], τ = tT ∈ [0,1] is the normalized movement duration, andϑ =− ln2ln tb , tb ∈]0,1[ is the movement time when the bounce posture is applied. We discretize t ∈ [0,T ] by NT equallyspaced points ti = iΔ, where Δ= TNT is the step size and i= 0,1, . . . ,NT . Our convention is that θ(ti,θ f ,θ b) representsθ(t,θ f ,θ b) at time ti. The two subproblems (Pa) and (Pb) are:(Pa) minθ f∈R1111∑k=1λk(θ0,k−θ f ,k)2,λk ≥ 0 (2)s.t. H(θ f )+dHO(θ f ,9) zˆ7(θ f )−Xtar = 0 (3)c(θ f ) = 0 (4)h f (θ f )≤ 0 (5)θmin ≤ θ f ≤ θmax (6)θ f ,8 = 0,θ f ,9 = θ f ,10 = θ f ,11 (7)(Pb) minθb∈R1111∑k=1λk(θ0,k−θb,k)2,λk ≥ 0 (8)s.t. θmin ≤ θ(ti,θ f ,θ b)≤ θmax (9)θ8(ti,θ f ,θ b) = 0,θ9(ti,θ f ,θ b) = θ10(ti,θ f ,θ b) (10)hb(θ(ti,θ f ,θ b))≤ 0, ti = 0, . . . ,T (11)θmin ≤ θ b ≤ θmax (12)where θmin,θmax are constant vectors that represent the lower and upper mechanical joint limits, Xtar ∈ R3 is theposition of the object that the robot must grasp, dHO(θ f ,9) is the distance from the center of the object to the palm ofthe hand. The equality constraints (4) depend on how one wants to grasp the object, i.e., the desired grip type. Theﬁngers joint angles (7) are determined using object dimensions and grip type. The nonlinear inequality constraintsdeﬁned by (5) and (11) are due to obstacle avoidance and were presented in [4]. The dimension of (Pa) dependson the number of constraints in (5), which depend on the number of obstacles in the workspace of the robot. Thisis, in general, a small-scale optimization problem (see Table 1). On the other hand, (Pb) is in general a large-scaleproblem (see Table 2), with the number of constraints in (11) depending on the number of obstacles and also on thetime discretization step that is used. Here we have introduced some changes to the model presented in [4] intended toimprove the modeling of reach to grasp movements. The ﬁrst concerns constraints (4) relative to hand orientation. Wehave replaced this equality by the inequality constraint ‖ c(θ f ) ‖2≤ δ , δ > 0. In fact, when grasping an object there isa higher priority in hand position (given by (3)) then on hand orientation, since it does not need to be meet perfectly.The second change concerns the constraints in (11), that are due to obstacle avoidance. Here we distinguish collisionof the robotic arm and hand with objects in the workspace of the robot and the collision with the object that is goingto be grasped. Thus, hb =(hb,hb)where hb ≡ hb(ε(t)) concerns the target object and hb concerns other objects inthe workspace. The function ε(t) is a monotonously decreasing to zero function representing the clearance distance.780RESULTSTables 1 and 2 report the results for four problems2, each consisting of two subproblems. These problems arisein tasks in which the anthropomorphic robot ARoS has to grasp objects using different grip types, and where thereach-to-grasp movement may be challenged by the presence of multiple obstacles. Speciﬁcally, we focus here onreaching and grasping columns and wheels, using a side and an above grip, respectively. For a deﬁnition of thegrip types and on the details of the task see e.g. [2, 3]. Previously in [4] we have tested the performance of threenonlinear constrained optimization solvers (IPOPT, KNITRO and SNOPT) for these four problems using a ﬁxeddiscretization of time and with sligtly different constraints (see previous section). All problems are coded in AMPL(http://www.ampl.com/) modelling language and were submitted to the NEOS Server3 platform. We report theresults for two nonlinear constrained optimization solvers, IPOPT and SNOPT, and four mixed integer nonlinearlyconstrained optimization solvers, BONMIN, COUENNE, FilMINT and MINLP. The choice on these solvers has todo with the fact they use different optimization techniques. IPOPT is an open source large-scale optimization solverthat implements an Interior Point (IP) ﬁlter line search method, while SNOPT implements an active-set SequentialQuadratic Programming (SQP) line search method. BONMIN is an open source software for the solution of mixedinteger nonlinearly constrained optimization problems that presents two classical algorithms: an outer-approximation-based branch-and-cut-based algorithm and a pure branch-and-bound algorithm. COUENNE solves the same typeof problems using a reformulation based branch-and-bound algorithm for a globally optimum solution. FilMINTcombines the branch-and-cut framework with ﬁlterSQP, an active set solver, used to solve the nonlinear programs thatarise as subproblems in the algorithm. MINLP implements a branch-and-bound algorithm and uses a ﬁlterSQP forsolving the continuous nonlinearly constrained optimization problems that arise as nodes of a tree. All these solversguarantee global optimal solutions only for convex mixed integer nonlinearly constrained optimization problems. OnlyCOUENNE guarantees global optimal solutions for nonconvex ones. For P1a and P2a the constraint (4) is given by‖ xˆ7(θ f )− zˆtar ‖2≤ δ , while for P3a and P4a is ‖ zˆ7(θ f )+ zˆtar ‖2≤ δ , where xˆtaryˆtarzˆtar is the local frame attachedto the object that the robot must grasp, and xˆ7 and zˆ7 are forward kinematics functions describing the orientation ofthe hand of the robot. In our implementation the value of the following constants are: T = 1, tb = 0.5, δ = 10−4and λk = 1,k = 1, . . . ,11. All solvers were run with the default options, with the exception of IPOPT, for which thesecond order derivatives information were approximated using a limited-memory Broyden-Fletcher-Goldfarb-Shanno(L-BFGS) method. Table 1 presents the numerical results for the small-scale problems P1a-P4a. For all problemsIPOPT, SNOPT, BONMIN and MINLP found an optimal solution. FiLMINT only found an optimal solution for P1aand P2a. COUENNE reported error for all problems. For all problems BONMIN found the worse solution. All theother solvers found the same solution for P2a, P3a and P4a. IPOPT found the best solution for P1a. FilMINT andMINLP found the same solution as SNOPT, which is not the best one. This may be explained by the fact that, for P1a,and since they are mixed integer nonlinearly constrained optimization solvers, they have incorrectly identiﬁed one ofthe variables as integer. In fact, for this problem all solvers, except IPOPT, found θ6 = 0. For P2a, P3a and P4a, noneof the variables is identiﬁed as integer. In terms of computational time, the best results were obtained by IPOPT andSNOPT.TABLE 1. Numerical results for the small-scale problems P1a-P4a.P1a(M = 45)P2a(M = 66)P3a(M = 45)P4a(M = 66)Obj. CPU(sec.) Obj. CPU(sec.) Obj. CPU(sec.) Obj. CPU(sec.)IPOPT 4.579811 0.071 4.579811 0.067 6.763172 0.049 6.763172 0.152SNOPT 5.913691 0.064 4.579814 0.631 6.763174 0.014 6.763176 0.017Bonmin 13.213600 1.025 14.114600 0.887 23.646300 1.090 8.342650 0.347FilMINT 5.913691 0.110 4.579814 0.170 † †MINLP 5.913691 ‡ 4.579814 ‡ 6.763176 ‡ 6.763176 ‡† No feasible solution found. ‡ Not available.M - number of constraints; Obj. - objective function value; CPU - Computational time in seconds.Table 2 reports the numerical results for the large-scale problems P1b-P4b for three different discretization steps(Δ = 0.1,0.05,0.025). As expected, the dimension of the problems, i.e., the number of constraints, M, and the CPU2 For the videos of all the trajectories see http://mobileanthropomorphicroboticsgroup.blogspot.com/.3 Note that, although we have access to the names of the machines used to solve the problems, we do not know their technical speciﬁcations.781TABLE 2. Numerical results for the large-scale problems P1b-P4b.P1b P2bΔ= 0.1 Δ= 0.05 Δ= 0.025 Δ= 0.1 Δ= 0.05 Δ= 0.025(M = 579) (M = 1199) (M = 2439) (M = 768) (M = 1598) (M = 3258)Obj./CPU (sec.) Obj./CPU (sec.) Obj./CPU (sec.) Obj./CPU (sec.) Obj./CPU (sec.) Obj./CPU (sec.)IPOPT 1.586542/1.560 0.727876/2.315 1.507861/27.674 1.586514/7.896 1.030850/3.978 2.180370/12.36#SNOPT 0.727859/0.078 0.727859/0.174 1.240994/0.668 0.932572/0.108 0.933127/0.312 1.621998/2.258P3b P4bIPOPT 0.765325/0.776 0.767077/1.259 0.768133/2.838 0.921092/3.039# 1.015694/5.015 1.44493/99.595#SNOPT 0.751773/0.061 0.754943/0.172 0.755340/0.422 0.834245/0.101 0.861077/0.236 0.837660/0.634# Converged to a point of local infeasibility; M - number of constraints; Obj. - objective function value; CPU - CPU in seconds.time increased as the time discretization step gets smaller. SNOPT presents a better performance both in terms of thevalue of the objective function and CPU time. In terms of the time step discretization, SNOPT presents the best resultsfor Δ= 0.1. This was not the case for IPOPT. In fact, for P1b and P2b, it presents the best results for a discretization ofΔ= 0.05. Furthermore, for these two problems and for Δ= 0.1, the trajectory found by IPOPT, results in the collisionof the ﬁngers of the robot, between the last two steps of the movement, with the object it is grasping4. For almost allproblems, the mixed integer nonlinearly constrained optimization solvers were not able to ﬁnd a solution. BONMINreported that the limit number of iterations has been reached for all problems except for P1b (Δ= 0.025), P2b (Δ= 0.1)and P4b (Δ= 0.1), in which the solution found was worse than the one found by IPOPT and SNOPT (23.460, 5.95432and 506.927, respectively). COUENNE reported error for all problems. FilMINT and MINLP reported failure inmemory allocation for all problems, except for P3b (Δ= 0.05), in which FilMINT found the same solution as IPOPT.CONCLUSIONS AND FUTURE WORKWe have solved the problem of generating human-like reaching and grasping movements of an anthropomorphic armand hand in the presence of obstacles, using different time discretization steps and different optimization techniques.IPOPT and SNOPT found the solutions for almost all problems and took less computational time. In general differentmethods found different solutions. The discretization of time still posses a problem. If it is too small the dimension ofthe problem increases greatly. On the other hand, if it is too large, collisions may occur. We tested the global solverCOUENNE, however it reported error. In the future we expect to do further tests with the global solver COUENNEand other global solvers that accept GAMS input.REFERENCES1. D. Rosenbaum, R. Meulenbroek, J. Vaugham, and C. Jansen, Posture-based Motion planning: Applications to grasping,Psychological Review 108, 709–734 (2001).2. E. Bicho, L. Louro, N. Hipólito, and W. Erlhagen, “A dynamic ﬁeld approach to goal inference and error monitoring forhuman-robot interaction,” in AISB Convention 2009 on Adaptive & Emergent Behaviour & Complex Systems: proc. of the Int.Symposium on New Frontiers in Human-Robot Interaction, edited by E. Dautenhahn, Edinburgh, Scotland, 2009, pp. 31–37.3. E. Bicho, W. Erlhagen, L. Louro, and E. Costa e Silva, Neuro-cognitive mechanisms of decision making in joint action: aHuman-Robot interaction study, Human Movement Science (January 3, 2011) doi:10.1016/j.humov.2010.08.012.4. E. Costa e Silva, F. Costa, E. Bicho, and W. Erlhagen, “Nonlinear Optimization for Human-like Movements of a High Degreeof Freedom Robotics Arm-hand System,” in 11th Int. Conf. on Computational Science and Applications (ICCSA 2011), editedby B. Murgante et al., Springer-Verlag, 2011, vol. 6784, Part III of Lecture Notes in Computer Science, pp. 327–342.5. U. Pattacini, F. Nori, L. Natale, G. Metta, and G. Sandini, “An experimental evaluation of a novel minimum-jerk cartesiancontroller for humanoid robots,” in Int. Conf. on Intelligent Robots and Systems (IROS2010), 2010, pp. 1668–1674.6. A. Wächter, and L. Biegler, On the implementation of an interior-point ﬁlter line-search algorithm for large-scale nonlinearprogramming, Mathematical Programming 106, 25–57 (2007).4 See the videos in http://mobileanthropomorphicroboticsgroup.blogspot.com/.782

Human-like movement of an anthropomorphic robot: problem revisited

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Human-like movement of an anthropomorphic robot: problem revisited

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