differential dynamic programming derivation

This paper presents a new balance control framework that combines constrained optimal control strategies with recent variational-based linearization approaches to solve the balancing problem for a common simplified quadruped model. Further, a Relaxed Barrier (ReB) method is used to manage inequality constraints and is integrated into HS-DDP for locomotion planning. Figure, 7 also compares the switching times obtained via the STO, algorithm with the algorithm proposed in [, feedback control is not used. Signal Temporal Logic (STL) has gained popularity in recent years as a specification language for cyber-physical systems, especially in robotics. Our trajectory optimization framework enables wheeled quadrupedal robots to drive over challenging terrain, e.g., steps, slopes, stairs, while negotiating these obstacles with dynamic motions. Whereas QP-based OSC only, considers the instantaneous effects of joint torques, whole-, body motion planning finds a sequence of torques by solving, a finite-horizon trajectory optimization (TO) problem, poten-, tially enabling recovery from larger disturbances. to the Differential Dynamic Programming (DDP) algorithm for trajectory, optimization in hybrid systems. Specifically, HS-DDP incorporates three algorithmic advances: an impact-aware DDP step addressing the impact event in legged locomotion, an Augmented Lagrangian (AL) method dealing with the switching constraint, and a Switching Time Optimization (STO) algorithm that optimizes switching times by leveraging the structure of DDP. The forward-backward, process above is repeated until the algorithm conv, This section presents a hybrid system model for bounding, quadrupeds. However it remains time consuming, whether using finite differences or automatic differentiation. Recent results (e.g., [1]) using Differential Dynamic Programming (DDP) [15] have shown great promise for online use of whole-body MPC. As opposed to a conventional Model-Predictive Control (MPC) approach that formulates a hierarchy of optimization problems, the proposed work formulates a single optimization problem posed over a hierarchy of models, and is thus named Model Hierarchy Predictive Control (MHPC). dynamic programming (HDP) algorithm is proven in the case of ... Morimoto et al. Therefore, contact locations, sequences and timings are not prespecified but optimized by the solver. Figure 5 compares the bounding gaits that are generated, by three methods: 1) A heuristic controller that is used to, warm start the optimization, 2) DDP (with impact-aware value, function update) that ignores switching constraints, and 3) AL, that enforces switching constraints. Details about DYNAMIC PROGRAMMING AND PARTIAL DIFFERENTIAL EQUATIONS, By Angel . Figure 8 explains why the two-level optimization strate, (28), (29), and (30), it is reasonable to update the control using, (8) and the switching times using (31) simultaneously since, the gradient and Hessian information are all available in the, than zero and is close to the predicted cost reduction, then. In, random samplingtechniquesareproposedtoimprovethescalabilityofDDP. The, second task applies the HS-DDP to quadruped bounding for, one gait cycle and demonstrates the efficiency of the ST, In this task, Algorithm 1 is applied to 2D quadruped, bounding for five gait cycles. We demonstrate the effectiveness of AL and ReB for switching constraints, friction constraints, and torque limits. Since DDP is a shooting method, the, algorithm can also be terminated at any time while still giving, Despite these benefits and promise, there are some difficul-, ties for DDP to be used in legged locomotion planning, such, as dealing with the impact discontinuity and managing various, constraints. Differential Dynamic Programming book. All the algorithms are implemented in our open-source C++ framework called Pinocchio. This paper presents a Differential Dynamic Programming (DDP) framework for trajectory optimization (TO) of hybrid systems with state-based switching. The entire framework is presented to, plan trajectories for the quadruped bounding model introduced, running cost and the terminal cost for the, body TO, the minimization of (16) is subject to the full-order, dynamics (14) and other various constraints. Dynamic Programming vs Divide & Conquer vs Greedy. Abstract: We present a hybrid differential dynamic programming (DDP) algorithm for closed-loop execution of manipulation primitives with frictional contact switches. ���lNC��}����r K�䳽=\98�� ���H��;|�qK��=��׿�ݙ�߰g�i�R�z���,�ΌII{PH$:�|]~���q1}˞Sk:���� )T���F�6�~wrT#����Ղ`̩���L����SM�QRN�.�Ps���M�]-Hې*�M�Wr�=��닲�����U:��lq�O����>� Numerous prior studies solve such a class of large non-convex optimal control problems in a hierarchical fashion. Unfortunately, the associated cost function is nonsmooth and non-convex. quadruped bounding. Widely used reduced-order models, include the Linear Inverted Pendulum (LIP) [, foothold locations, with the Zero-Moment Point (ZMP) cri-, terion used to enforce admissible CoM trajectories [, This paper was recommended for publication by Editor Dezhen Song upon, evaluation of the Associate Editor and Revie, Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556. Despite that, DDP has been associated with poor numerical convergence, particularly when considering long time horizons. Different from the previ-, ous task, where only the control is optimized, switching times, are also optimized in this task. Quadratic running cost and terminal cost are used in (16), and energy consumption in running cost, respectively, is the weighting matrix for the terminal cost (of the. model. ����ԡ��B+`��耙�� 6�A�M�d�B������U�2��pie 6�}� �4����C!S/� K"���+S'C3O�����l�s.2�f.��cbn�dx�`Ƽ��{u �����2�21{1�3��;���Q�u �c;�{� l��Z��x�g���܏�t"ϊ���n Na����3 L�� endstream endobj 537 0 obj <> endobj 538 0 obj <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI]/XObject<>>>/Rotate 0/Type/Page>> endobj 539 0 obj <>stream Intuitively, the HJB equation can be derived as follows. Overview of the HS-DDP algorithmic framework. Section V an-, alyzes the performance of the proposed algorithm in terms, of constraint handling and efficiency of the STO as applied to. Its scalability, fast convergence rate, and feedback control The paper simply shows H as a one element matrix with x2+x3 inside, but it doesn't show any derivation for the sake of brevity. The performance of the developed algorithms is benchmarked on a simulation model of the MIT Mini Cheetah executing a bounding gait. Suppose that, here for simplicity. While Algorithm 1 finds the optimal control, optimization (STO) algorithm developed in this section up-, The STO algorithm reformulates the OCP (16) on fixed, time intervals of length one, and augments the state vector. Similar results are observed for the back leg. Model Hierarchy Predictive Control of Robotic Systems, Trajectory Optimization for High-Dimensional Nonlinear Systems under STL Specifications, Dynamic Locomotion in the MIT Cheetah 3 Through Convex Model-Predictive Control, Differential Dynamic Programming for Multi-Phase Rigid Contact Dynamics, Fast Online Trajectory Optimization for the Bipedal Robot Cassie, Analytical Derivatives of Rigid Body Dynamics Algorithms, Whole-Body Nonlinear Model Predictive Control Through Contacts for Quadrupeds, Feedback MPC for Torque-Controlled Legged Robots, ALTRO: A Fast Solver for Constrained Trajectory Optimization, Fast Direct Multiple Shooting Algorithms for Optimal Robot Control, Mini Cheetah: A Platform for Pushing the Limits of Dynamic Quadruped Control, Dynamic Locomotion Through Online Nonlinear Motion Optimization for Quadrupedal Robots, Hybrid Systems Differential Dynamic Programming for Whole-Body Motion Planning of Legged Robots, Trajectory Optimization for Wheeled-Legged Quadrupedal Robots Driving in Challenging Terrain. Then, Algorithm 1 can, are then used to update the switching times, Cost function (30), scaled dynamics (28) and switching, Execute Algorithm 1 to obtain optimal control, on optimality is reduced by including the, are weighting matrices for state deviation, ms is used, and the switching times are selected, The International Journal of Robotics Resear, IEEE transactions on systems, man, and cybernetics. This allows for gradient based optimization of parameters in the program, often via gradient descent.Differentiable programming has found use in a wide variety of areas, particularly scientific computing and artificial intelligence. DDP background, and the hybrid dynamics formulation are given in Sections II, and III. Differential Dynamic Programming Neural Optimizer. Note that the transition from stance to flight is continuous, the state-switched hybrid system, (9) and (10) are equivalent. A common strategy to generate efficient locomo-tion movements is to split the problem into two consecutive steps: the first one generates the contact sequence together with the centroidal trajectory, while the second step computes the whole-body trajectory that follows the centroidal pattern. The update, distinction, one execution of the forward sweep and backward, sweep of DDP is called one DDP iteration. The one-gait-cycle bounding example compares the developed, STO algorithm to the previous solutions, demonstrating that, our method is more efficient due to the inclusion of the, Though forward Euler integration is used in this work for, dynamics simulation, the developed HS-DDP is independent, of the integration scheme. Other previous work has contributed to at-, tacking the second difficulty by leveraging constraint-handling, solving QPs with a Projected Newton algorithm in [, that results when penalty coefficients are large. It is closely related to Pantoja's step-wise Newton's method. Since then, it has found ap-plications in many complex, high-dimensional, engineering problems (see, for example, [15], [16], [17], [18], [19]). Using the chain rule and adequate algebraic differentiation of spatial algebra, we firstly differentiate explicitly RNEA. Pseudocode for. The ReB algorithm is executed whenever the AL, algorithm is executed. f� �s��j��s�ׯ�J)J�{R��v�^��)�N)��S����6�v.5rk)+��1��"4T�3)�M The AL algorithm terminates, when all switching constraints are satisfied. Nevertheless, it can be combined with various, constraint-handling techniques from NLP for constrained op-, timization. The, mobility afforded by legged robots makes them exceptionally, suitable for these scenarios. Sequential snapshots of the generated bounding motion for Mini Cheetah. Differential Dynamic Programming (DDP) [1] is a well-known trajectory optimization method that iteratively finds a locally optimal control policy starting from a nominal con-trol and state trajectory. Yet, when attempting to transition these solutions to balance in legged robots, technical, This work presents a successful implementation of a nonlinear optimization-based Regularized Predictive Control (RPC) for legged locomotion on the MIT Cheetah 3 robot platform. The obtained results underline the performance, transferability and robustness of the approach. In this section, we are particularly interested in the, switching equality constraint (17d). Sometimes, this doesn't optimise for the whole problem. ReB method. the optimal switching times alongside the optimal control. Given a nominal control, sequence, the forward sweep computes a nominal trajectory, is then executed to generate a policy that is used to update, the control sequence. details that make the theory work in practice and many times it is precisely these details that end up being critical to the success or failure of the theory in real world applications. This results in modest memory requirements for its defining parameters and rapid convergence. Bottom: Joint torques. Overall, these approaches have the advantage of f, respectively denote the state and control, the value function (i.e., optimal cost-to-go), is rarely possible due to nonlinearity of, denotes one gait cycle. The terrain map, together with the use of a stability constraint, allows the optimizer to generate feasible motions that cannot be discovered without taking the terrain information into account. Bottom: Motion generated with AL enforcing switching constraints. The obtained … Despite the, appeal of this approach, the curse of dimensionality caused, by the high-dimensional state space of legged robots has, using Differential Dynamic Programming (DDP) [, shown great promise for online use. Consider the discrete-time optimal control problem min U J(X;U) = min U P N 1 k=0 l(x k;u k) + ˚(x N) subject to: x k+1 = f(x k;u k); k= 0;:::;N 1: (1) where x k 2Rn, u The algorithm uses locally-quadratic models of the dynamics and cost functions, and displays quadratic convergence. Since its introduction in [1], there has been a plethora of variations and applications of DDP within the controls and robotics communities. Omitting the third terms in the last three, equations gives rise to the iLQR algorithm, which enables, faster iterations but loses quadratic conv, employ iLQR in this work and use the algorithm proposed in, stituting (6) into the equation (4) results in update equations, The equations (5) and (7) are computed recursively starting, at the final state, constituting the backward pass of DDP, nominal control is then updated using the resulting control, respectively are the nominal and new state-control pair, backtracking line search method is used to select, decrease of the cost in each iteration. 536 0 obj <> endobj 573 0 obj <>/Filter/FlateDecode/ID[<9DF7E84DDE21292641D5A25714E21C1E><51B5232D83684FE1AA946E89A4F4E982>]/Index[536 112]/Info 535 0 R/Length 152/Prev 836201/Root 537 0 R/Size 648/Type/XRef/W[1 3 1]>>stream The gen-, eralized coordinates for this 2D quadruped are, anymore, and the KKT matrix degenerates to the inertia matrix, multiplying both sides of (11) by the inverse of the KKT, matrix and separating out the solution for, While the generalized coordinates remain unchanged across, impact events, velocities change instantaneously at each, that the contact foot sticks to the ground after impact. This study investigates an approach of Alternating Direction Method of Multipliers (ADMM) and proposes a new splitting scheme for legged locomotion problems. the cost function, avoiding the numerical ill-conditioning. Given any inequality, constrained optimization problem as below, ReB attacks (25) by successively solving the unconstrained, method allows the objective function to be e, infeasible trajectory, which cannot be done with a standard, loop. With AL and ReB, the non-negativity of normal GRF, friction, and torque limit constraints are satisfied in four AL iterations. A ReB method is combined with HS-DDP to manage the, inequality constraints. This paper presents a new predictive control architecture for high-dimensional robotic systems. More details can be found in [5, 10]. Accelerated ADMM based Trajectory Optimization for Legged Locomotion with Coupled Rigid Body Dynamic... A Direct-Indirect Hybridization Approach to Control-Limited DDP, Variational-Based Optimal Control of Underactuated Balancing for Dynamic Quadrupeds. Our framework has been tested in simulation and on ANYmal, a fully torque-controllable quadrupedal robot, both in simulation and on the actual robot. t due to small changes in state; variables instead of the cost itself'. C�1^P����DrAWL"��+S�9�UI���i�!$�`���DZ̗F�N��Ho�Jd0 �(n�n%�A�&J:����]���+!Q�H+x�,g�H��ĭ���n�DL�b�ZH���KhDH��a��ȴ3�ƑQ�t%��>Kz One of the main reasons is due to system instabilities and poor warm-starting (only controls). In this paper, we propose new algorithms to efficiently compute them thanks to closed-form formulations. In this simulation, we have zero penalty on forward position. The major, difference between our method and the approach in [, inclusion of this feedback term in the control law, policy used in this work allows (31) to make more aggressi, updates, and consequently achieves faster con, perturbations to the locally optimal trajectory. GRF and joint toques for 2D Mini Cheetah bounding. The trajectories have been experimentally verified on quadrupedal robot ANYmal equipped with non-steerable torque-controlled wheels. As a result, existing synthesis methods scale poorly to high-dimensional nonlinear systems. This algorithm continuously calls, violations are remeasured and added to the cost function, and, another DDP call is executed. The integration time, such that the flight mode (and the front-stance mode) runs, for 72 ms and the back stance mode runs for 80 ms. DDP and presents the ReB method for inequality constraints. This paper presents control-limited Feasibility-driven DDP (Box-FDDP), a solver that incorporates a direct-indirect hybridization of the control-limited DDP algorithm. Differential Dynamic Programming (DDP) is an indirect method which optimizes only over the unconstrained control-space and is therefore fast enough to allow real-time control of a full hu- manoid robot on modern computers. The DDP algorithm, introduced in, computes a quadratic approximation of the cost-to-go and correspondingly, a local linear-feedback controller. In this work, this time-switched reformulation is considered, This section discusses three algorithmic advances for HS-. Lantoine et al. constraint violation after the total cost is converged. are optimized under switching constraints. The controller is implemented as a convex quadratic program (QP) that uses an unconstrained optimal control solution to approximate a friction-constrained optimal policy. Augmented, adding a linear multiplier term. The resulting algorithm, known as Stochastic Differential Dynamic Programming (SDDP), is a generalization of iLQG. The developed algorithms are. All figure content in this area was uploaded by He Li, All content in this area was uploaded by He Li on Jul 16, 2020, Hybrid Systems Differential Dynamic Programming, for Whole-Body Motion Planning of Legged Robots, gramming (DDP) framework for trajectory optimization (TO), of hybrid systems with state-based switching. information while computing the plans. Planning and control of these primitives is challenging as they are hybrid, under-actuated, and stochastic. In this work we present a whole-body Nonlinear Model Predictive Control approach for Rigid Body Systems subject to contacts. Via comparison to model-predictive control strategies, the proposed formulation is highly compact, requiring less computation, while still showing the ability to handle extreme friction limitations. The algorithm reduces the, time of the first flight mode and the front-stance mode. of motion plans, and the management of various constraints. In particular, the methods focus on handling, the impact event, the associated switching constraints, and the inequality. the AL algorithm is shown in Algorithm 1. manage the inequality constraints in (17). CONCLUSION The main purpose of this work has been to present some exact expressions for the change of cost due to arbitrary controls, and to exhibit the central role these expressions 248 DIFFERENTIAL DYNAMIC PROGRAMMING can play, both in control theory and numerical optimization, by illustrating their application to the derivation of algorithms and conditions of … We prove the soundness of our proposed approach, demonstrate order-of-magnitude speed improvements over the state-of-the-art on several benchmark problems, and demonstrate the scalability of our approach to the full nonlinear dynamics of a 7 degree-of-freedom robot arm. the effectiveness of ReB for enforcing inequality constraints. Interested readers may refer to [. The ef, feedback term in control to account for perturbations. We evaluate and validate the performance of the proposed ADMM algorithm on a car-parking example and a bipedal locomotion problem over rough terrains. The … Hardware experiments in form of periodic and non-periodic tasks are applied to two quadrupeds with different actuation systems.

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