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144 nips-2002-Minimax Differential Dynamic Programming: An Application to Robust Biped Walking

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Author: Jun Morimoto, Christopher G. Atkeson

Abstract: We developed a robust control policy design method in high-dimensional state space by using differential dynamic programming with a minimax criterion. As an example, we applied our method to a simulated ﬁve link biped robot. The results show lower joint torques from the optimal control policy compared to a hand-tuned PD servo controller. Results also show that the simulated biped robot can successfully walk with unknown disturbances that cause controllers generated by standard differential dynamic programming and the hand-tuned PD servo to fail. Learning to compensate for modeling error and previously unknown disturbances in conjunction with robust control design is also demonstrated.

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We developed a robust control policy design method in high-dimensional state space by using differential dynamic programming with a minimax criterion. [sent-7, score-0.635]

2 As an example, we applied our method to a simulated ﬁve link biped robot. [sent-8, score-0.301]

3 The results show lower joint torques from the optimal control policy compared to a hand-tuned PD servo controller. [sent-9, score-0.344]

4 Results also show that the simulated biped robot can successfully walk with unknown disturbances that cause controllers generated by standard differential dynamic programming and the hand-tuned PD servo to fail. [sent-10, score-0.943]

5 Learning to compensate for modeling error and previously unknown disturbances in conjunction with robust control design is also demonstrated. [sent-11, score-0.315]

6 1 Introduction Reinforcement learning[8] is widely studied because of its promise to automatically generate controllers for difﬁcult tasks from attempts to do the task. [sent-12, score-0.043]

7 However, reinforcement learning requires a great deal of training data and computational resources, and sometimes fails to learn high dimensional tasks. [sent-13, score-0.034]

8 To improve reinforcement learning, we propose using differential dynamic programming (DDP) which is a second order local trajectory optimization method to generate locally optimal plans and local models of the value function[2, 4]. [sent-14, score-0.278]

9 However, when we apply dynamic programming to a real environment, handling inevitable modeling errors is crucial. [sent-16, score-0.11]

10 In this study, we develop minimax differential dynamic programming which provides robust nonlinear controller designs based on the idea of H∞ control[9, 5] or risk sensitive control[6, 1]. [sent-17, score-0.65]

11 We apply the proposed method to a simulated ﬁve link biped robot (Fig. [sent-18, score-0.434]

12 Our strategy is to use minimax DDP to ﬁnd both a low torque biped walk and a policy or control law to handle deviations from the optimized trajectory. [sent-20, score-0.799]

13 We show that both standard DDP and minimax DDP can ﬁnd a local policy for lower torque biped walk than a handtuned PD servo controller. [sent-21, score-0.87]

14 We show that minimax DDP can cope with larger modeling error than standard DDP or the hand-tuned PD controller. [sent-22, score-0.298]

15 Thus, the robust controller allows us to collect useful training data. [sent-23, score-0.281]

16 In addition, we can use learning to correct modeling ∗ also afﬁliated with Human Information Science Laboratories, Department 3, ATR International errors and model previously unknown disturbances, and design a new more optimal robust controller using additional iterations of minimax DDP. [sent-24, score-0.579]

17 Differential dynamic programming maintains a second order local model of a Q function (Q(i), Qx (i), Qu (i), Qxx (i), Qxu (i), Quu (i)), where Q(i) = r(xi , ui , i) + V (xi+1 , i + 1), and the subscripts indicate partial derivatives. [sent-27, score-0.354]

18 Then, we can derive the new control output unew = ui + δui from arg maxδui Q(xi + δxi , ui + δui , i). [sent-28, score-0.608]

19 Finally, i by using the new control output unew , a second order local model of the value function i (V (i), Vx (i), Vxx (i)) can be derived [2, 4]. [sent-29, score-0.138]

20 2 Finding a local policy DDP ﬁnds a locally optimal trajectory xopt and the corresponding control trajectory uopt . [sent-31, score-0.347]

21 i i When we apply our control algorithm to a real environment, we usually need a feedback controller to cope with unknown disturbances or modeling errors. [sent-32, score-0.495]

22 Fortunately, DDP provides us a local policy along the optimized trajectory: uopt (xi , i) = uopt + Ki (xi − xopt ), i i (2) where Ki is a time dependent gain matrix given by taking the derivative of the optimal policy with respect to the state [2, 4]. [sent-33, score-0.28]

23 The difference is that the proposed method has an additional disturbance variable w to explicitly represent the existence of disturbances. [sent-36, score-0.097]

24 This representation of the disturbance provides the robustness for optimized trajectories and policies [5]. [sent-37, score-0.204]

25 Here, δui and δwi must be chosen to minimize and maximize the second order expansion of the Q function Q(xi + δxi , ui + δui , wi + δwi , i) in (3) respectively, i. [sent-39, score-0.357]

26 , δui = −Q−1 (i)[Qux (i)δxi + Quw (i)δwi + Qu (i)] uu δwi = −Q−1 (i)[Qwx (i)δxi + Qwu (i)δui + Qw (i)]. [sent-41, score-0.038]

27 ww (13) By solving (13), we can derive both δui and δwi . [sent-42, score-0.04]

28 1 Biped robot model In this paper, we use a simulated ﬁve link biped robot (Fig. [sent-45, score-0.567]

29 Kinematic and dynamic parameters of the simulated robot are chosen to match those of a biped robot we are currently developing (Fig. [sent-47, score-0.583]

30 Height and total weight of the robot are about 0. [sent-49, score-0.133]

31 3 link3 joint2,3 4 link4 link2 joint4 2 link1 joint1 5 1 link5 ankle Figure 1: Left: Five link robot model, Right: Real robot Table 1: Physical parameters of the robot model link1 link2 link3 link4 link5 mass [kg] 0. [sent-53, score-0.528]

32 75 We can represent the forward dynamics of the biped robot as xi+1 = f (xi ) + b(xi )ui , (15) ˙ ˙ where x = {θ1 , . [sent-68, score-0.437]

33 , θ5 } denotes the input state vector, u = {τ1 , . [sent-74, score-0.079]

34 , τ4 } denotes the control command (each torque τj is applied to joint j (Fig. [sent-77, score-0.249]

35 In the minimax optimization case, we explicitly represent the existence of the disturbance as xi+1 = f (xi ) + b(xi )ui + bw (xi )wi , (16) where w = {w0 , w1 , w2 , w3 , w4 } denotes the disturbance (w0 is applied to ankle, and wj (j = 1 . [sent-79, score-0.499]

36 The term (xi − xd )T Q(xi − xd ) encourages the robot to i i follow the nominal trajectory, the term ui T Rui discourages using large control outputs, and the term (v(xi ) − v d )T S(v(xi ) − v d ) encourages the robot to achieve the desired velocity. [sent-87, score-0.735]

37 (19) The term F (x0 ) penalizes an initial state where the foot is not on the ground: F (x0 ) = Fh T (x0 )P0 Fh (x0 ), (20) where Fh (x0 ) denotes height of the swing foot at the initial state x0 . [sent-89, score-0.201]

38 The term ΦN (x0 , xN ) is used to generate periodic trajectories: ΦN (x0 , xN ) = (xN − H(x0 ))T PN (xN − H(x0 )), (21) where xN denotes the terminal state, x0 denotes the initial state, and the term (xN − H(x0 ))T PN (xN − H(x0 )) is a measure of terminal control accuracy. [sent-90, score-0.249]

39 A function H() represents the coordinate change caused by the exchange of a support leg and a swing leg, and the velocity change caused by a swing foot touching the ground (Appendix A). [sent-91, score-0.211]

40 We implement the minimax DDP by adding a minimax term to the criterion. [sent-92, score-0.504]

41 We use a modiﬁed objective function: N −1 wi T Gwi , Jminimax = J − (22) i=0 where wi denotes a disturbance vector at the i-th time step, and the term wi T Gwi rewards coping with large disturbances. [sent-93, score-0.504]

42 This explicit representation of the disturbance w provides the robustness for the controller [5]. [sent-94, score-0.328]

43 4 Results We compare the optimized controller with a hand-tuned PD servo controller, which also is the source of the initial and nominal trajectories in the optimization process. [sent-95, score-0.528]

44 0} in equation (21), where IN denotes N dimensional identity matrix. [sent-111, score-0.053]

45 For minimax DDP, we set the parameter for the disturbance reward in equation (22) as G = diag{5. [sent-112, score-0.349]

46 0} (G with smaller elements generates more conservative but robust trajectories). [sent-117, score-0.048]

47 Each parameter is set to acquire the best results in terms of both the robustness and the energy efﬁciency. [sent-118, score-0.023]

48 When we apply the controllers acquired by standard DDP and minimax DDP to the biped walk, we adopt a local policy which we introduced in section 2. [sent-119, score-0.616]

49 Results in table 2 show that the controller generated by standard DDP and minimax DDP did almost halve the cost of the trajectory, as compared to that of the original hand-tuned PD servo controller. [sent-121, score-0.697]

50 However, because the minimax DDP is more conservative in taking advantage of the plant dynamics, it has a slightly higher control cost than the standard DDP. [sent-122, score-0.37]

51 N −1 1 Note that we deﬁned the control cost as N i=0 ||ui ||2 , where ui is the control output (torque) vector at i-th time step, and N denotes total time step for one step trajectories. [sent-123, score-0.544]

52 Table 2: One step control cost (average over 100 steps) PD servo standard DDP minimax DDP control cost [(N · m)2 × 10−2 ] 7. [sent-124, score-0.683]

53 86 To test robustness, we assume that there is unknown viscous friction at each joint: dist ˙ τj = −µj θj (j = 1, . [sent-127, score-0.069]

54 , 4), (23) where µj denotes the viscous friction coefﬁcient at joint j. [sent-130, score-0.128]

55 We used two levels of disturbances in the simulation, with the higher level being 3 times larger than the base level (Table 3). [sent-131, score-0.128]

56 Table 3: Parameters of the disturbance µ2 ,µ3 (hip joints) µ1 ,µ4 (knee joints) base 0. [sent-132, score-0.118]

57 15 All methods could handle the base level disturbances. [sent-136, score-0.021]

58 Both the standard and the minimax DDP generated much less control cost than the hand-tuned PD servo controller (Table 4). [sent-137, score-0.754]

59 However, only the minimax DDP control design could cope with the higher level of disturbances. [sent-138, score-0.389]

60 Figure 2 shows trajectories for the three different methods. [sent-139, score-0.065]

61 Both the simulated robot with the standard DDP and the hand-tuned PD servo controller fell down before achieving 100 steps. [sent-140, score-0.55]

62 The bottom of ﬁgure 2 shows part of a successful biped walking trajectory of the robot with the minimax DDP. [sent-141, score-0.742]

63 Figure 3 shows ankle joint trajectories for the three different methods. [sent-142, score-0.2]

64 Only the minimax DDP successfully kept ankle joint θ 1 around 90 degrees more than 20 seconds. [sent-143, score-0.387]

65 Table 5 shows the number of steps before the robot fell down. [sent-144, score-0.152]

66 We terminated a trial when the robot achieved 1000 steps. [sent-145, score-0.133]

67 Table 4: One step control cost with the base setting (averaged over 100 steps) PD servo standard DDP minimax DDP control cost [(N · m)2 × 10−2 ] 8. [sent-146, score-0.704]

68 87 Hand-tuned PD servo Standard DDP Minimax DDP Figure 2: Biped walk trajectories with the three different methods 5 Learning the unmodeled dynamics In section 4, we veriﬁed that minimax DDP could generate robust biped trajectories and policies. [sent-149, score-0.951]

69 The minimax DDP coped with larger disturbances than the standard DDP and the hand-tuned PD servo controller. [sent-150, score-0.535]

70 However, if there are modeling errors, using a robust controller which does not learn is not particularly energy efﬁcient. [sent-151, score-0.28]

71 Fortunately, with minimax DDP, we can collect sufﬁcient data to improve our dynamics model. [sent-152, score-0.334]

72 Here, we propose using Receptive Field Weighted Regression (RFWR) [7] to learn the error dynamics of the biped robot. [sent-153, score-0.304]

73 In this section we present results on learning a simulated modeling error (the disturbances discussed in section 4). [sent-154, score-0.164]

74 We can represent the full dynamics as the sum of the known dynamics and the error dynamics ∆F(xi , ui , i): xi+1 = F(xi , ui ) + ∆F(xi , ui , i). [sent-156, score-0.909]

75 We align 20 basis functions (N b = 20) at even intervals along the biped trajectories. [sent-158, score-0.247]

76 The learning strategy uses the following sequence: 1) Design the initial controller using minimax DDP applied to the nominal model. [sent-159, score-0.52]

77 4) Redesign the biped controller using minimax DDP with the learned model. [sent-162, score-0.707]

78 Results in table 6 show that the controller after learning the error dynamics used lower torque to produce stable biped walking trajectories. [sent-164, score-0.676]

79 Table 6: One step control cost with the large disturbances (averaged over 100 steps) without learned model with learned model control cost [(N · m)2 × 10−2 ] 17. [sent-165, score-0.362]

80 3 6 Discussion In this study, we developed an optimization method to generate biped walking trajectories by using differential dynamic programming (DDP). [sent-167, score-0.506]

81 We showed that 1) DDP and minimax DDP can be applied to high dimensional problems, 2) minimax DDP can design more robust controllers, and 3) learning can be used to reduce modeling error and unknown disturbances in the context of minimax DDP control design. [sent-168, score-1.071]

82 Both standard DDP and minimax DDP generated low torque biped trajectories. [sent-169, score-0.579]

83 We showed that the minimax DDP control design was more robust than the controller designed by standard DDP and the hand-tuned PD servo. [sent-170, score-0.623]

84 Given a robust controller, we could collect sufﬁcient data to learn the error dynamics using RFWR[7] without the robot falling down all the time. [sent-171, score-0.263]

85 We also showed that after learning the error dynamics, the biped robot could ﬁnd a lower torque trajectory. [sent-172, score-0.46]

86 DDP provides a feedback controller which is important in coping with unknown distur- bances and modeling errors. [sent-173, score-0.298]

87 However, as shown in equation (2), the feedback controller is indexed by time, and development of a time independent feedback controller is a future goal. [sent-174, score-0.464]

88 Appendix A Ground contact model The function H() in equation (21) includes the mapping (velocity change) caused by ground contact. [sent-175, score-0.115]

89 To derive the ﬁrst derivative of the value function V x (xN ) and the second derivative Vxx (xN ), where xN denotes the terminal state, the function H() should be analytical. [sent-176, score-0.08]

90 Rigid body collisions of planar kinematic chains with multiple contact points. [sent-195, score-0.084]

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