Discrete Mechanics And Optimal Control Research Articles

A Novel Superlinearly Convergent Trust Region-Sequential Quadratic Programming Approach for Optimal Gait of Bipedal Robots Via Nonlinear Model Predictive Control

A type of trust region-sequential quadratic programming (TRSQP) approach with superlinearly convergent property is first proposed, investigated, and implemented on bipedal robots based on nonlinear model predictive control (NMPC). NMPC is utilized to predict the system behavior and optimize the control move in a receding horizon way, which will result in recursive efficiency and stability. Considering that the classical line search rules are expensive or hard in particular applications, the attempted trust region search is leveraged to avoid the drawbacks of the classical line search rules. Moreover, the feasible descent direction is contained in the trust region via a novelly truncated technique which avoids to recompute the quadratic programming subproblem (QPS) for the main search direction. Owing to some suitable conditions, the globally/superlinearly convergent performance and well-defined properties are analyzed and verified for the TRSQP. The main result is illustrated on a simple bipedal robot which is called as compass-like bipedal robot (CLBR) through numerical simulations and is used to generate dynamic locomotion via TRSQP and NMPC. Furthermore, to demonstrate the feasibility and superiority of a complex bipedal robot which is called as a high-dimensional bipedal robot, numerical simulations are conducted on the model of a three-link bipedal robot (TLBR) and a five-link robot (RABBIT). Furthermore, simulation results show that the TRSQP approach is effectiveness and superiority through comparing with the classical approaches, which included discrete mechanics and optimal control (DMOC) and hybrid zero dynamic (HZD), and control Lyapunov function-quadratic programming (CLF-QP) for the optimal gait of bipedal robot. In addition, to verify the robustness, the TLBR model with parameter’s disturbance 1.5 times is investigated and analyzed via TRSQP with NMPC technique. Last,this study develops an interesting framework to exploiting control methods on bipedal robots through accurately and effectively solving nonlinear programming problems.

Variational Integrators and Optimal Control for a Hybrid Pendulum‐on‐Cart‐System

AbstractHybrid systems combine continuous‐time and discrete‐time dynamics. They emerge in mechanical systems, inter alia, when collisions are modelled by impulse‐like contact forces. During collisions of two or more bodies, the positions of the latter evolve continuously, but their velocities experience jumps. On the one hand, this non‐smoothness significantly complicates the optimal control problem; on the other, it opens up opportunities for new numerical methods. This contribution uses a nested optimization in order to control the swinging up of a hybrid pendulum‐on‐cart‐system. Both, pendulum and cart, move in finite ranges which are bounded by limiters (elastic collisions). The outer layer optimizes the occurence and the momenta of the collisions, while the inner layer optimizes the smooth motions from start to collision, from collision to collision, and finally to the terminal state. The discretization and the optimal control of continuous‐time dynamics are based on the Discrete‐Mechanics‐and‐Optimal‐Control (DMOC) approach.

Variational integrator based optimal feedback control for constrained mechanical systems

AbstractToday, a lot of mechanical systems have to operate with an improved performance compared to equal constructions decades ago. To stay competitive, engineers need to optimise all aspects of a mechanical system including its feedback control. An approach to minimize the control costs and ensuring a stable deviation control is the Riccati controller. To describe the discrete dynamics, a constrained variational integrator is used, which is a variant of a structure‐preserving integration scheme. The desired optimal trajectory and according control input is determined solving the discrete mechanics and optimal control (DMOC) algorithm based on the variational integrator. Then, during time stepping of the perturbed system, the discrete Riccati equations yield the optimal deviation control input. Adding the optimal control input to the deviation control value causes a structure preserving trajectory as both DMOC and Riccati equations are based on the same variational integrator. In this work, criteria for the choice of quadrature rules required to derive the variational integrator are investigated to close a gap of previous works in this field.This procedure is applied to three different coordinate choices, minimal, redundant and nullspace coordinates. Simulation examples show that a stable handling of highly‐nonlinear systems is assured. With applying the formulas derived in this work, a transformation between different coordinate parametrisations and the corresponding cost matrices leads to the same deviation control and thus to the same system behaviour. Nevertheless, it is in the responsibility of the application engineer to choose appropriate cost‐matrices for the occurring perturbations.

Fast Nonlinear Moving Horizon Estimation with Discrete Mechanics and Optimal Control

This paper proposes a novel sub-optimal nonlinear moving horizon estimation (NMHE) algorithm tailored to underactuated mechanical systems. We utilize the special variational structure of the mechanical systems and achieve numerical integration in NMHE via variational integrators. The resulting NMHE algorithm is based on the optimal control method Discrete Mechanics and Optimal Control (DMOC). It has significantly less number of optimization variables and is numerically more efficient than the standard NMHE algorithm based on multiple-shooting (MS). We use detailed numerical analysis with a double pendulum on a cart system to demonstrate that the novel NMHE algorithm outperforms the famous Extended Kalman Filter and is more efficient than the standard NMHE approach based on MS and the partial condensing technique in the chosen application. We also show closed-loop simulation results with the combined DMOC-based NMHE-NMPC framework for the swing-up and stabilization of the double pendulum system.

Variational integrator for constrained mechanical systems with pulsed disturbances and optimal feedback control

AbstractAn approach to minimize the control costs and ensuring a stable deviation control is the Riccati controller and we use it to control constrained dynamical systems (differential algebraic equations of Index 3) with pulsed disturbances. To describe their discrete dynamics, a constrained variational integrator [1] is used. Using a discrete version of the Lagrange‐d'Alembert principle yields a forced constrained discrete Euler‐Lagrange equation in a position‐momentum form that depends on the current and future time steps [2]. The desired optimal trajectory (qopt, popt) and according control input uopt is determined solving the discrete mechanics and optimal control (DMOC) algorithm [3] based on the variational integrator. Then, during time stepping of the perturbed system, the discrete Riccati equation yields the optimal deviation control input uR. Adding uopt and uR to the discrete Euler‐Lagrange equation causes a structure preserving trajectory as both DMOC and Riccati equations are based on the same variational integrator. Furthermore, coordinate transformations are implemented (minimal, redundant and nullspace) enabling the choice of different coordinates in the feedback loop and in the optimal control problem. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Optimal control of a slot car racer

AbstractIn this contribution, we present a simulation method for the optimal control of a mechatronic system that is based on discrete variational calculus and apply it to compute the time‐minimal path of a slot car racer. Here, the DMOC (Discrete Mechanics and Optimal Control [4]) method is used to generate offline optimal trajectories for the electro‐mechanically coupled system, i.e. sequences of discrete configurations and sequences of driving voltages. The time‐minimal path is achieved by the choice of different cost functions, the sum of time steps, the negative sum of the quadratic momenta and the negative sum of the quadratic velocities. Simulation results show that the momentum formulation yields the lowest number of iterations. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Towards Discrete Mechanics and Optimal Control for Complex Models * *Financial support by the European Commission within the H2020 project Spexor (GA 687662) is gratefully acknowledged

Abstract Methods for trajectory optimization originate from the need to solve spaceflight mechanics problems beginning in the 1950s and 1960s. The rise of digital compute power popularized so-called direct methods from the 1980s onwards. Since the 1990s, these methods have been used extensively in various fields like robotics and chemical reaction kinetics. Discrete Mechanics and Optimal Control (DMOC) is a relatively new but not very widespread method that offers nice mathematical properties. It has mostly been applied to small mechanical problems so far. We show that the method can offer competitive performance when being used on more complex models like a 2D human model. Furthermore, we point out that the method can take advantage of parallel compute architectures nicely. It relies on a variational principle and temporal discretization that lead to a sparse constraint Jacobian that scales well.

Efficient Planning of Optimal Trajectory for a Furuta Double Pendulum Using Discrete Mechanics and Optimal Control

This paper presents a more accurate model of a Furuta double pendulum for efficient planning of optimal trajectories using the new approach of discrete mechanics and optimal control (DMOC). Based on the variation of discrete mechanics a direct discretization of the Lagrange-d’Alembert principle enables a novel formulation of the optimization problem. This leads to less optimization parameters and thus more computational efficiency. To shed the light on the advantages of the proposed method the swing-up maneuver of the Furuta double pendulum is used to demonstrate the point-to-point trajectory planning. For this purpose, a model of Furuta double pendulum based on Denavit-Hartenberg convention with higher accuracy is presented additionally.

Optimal feedback control for constrained mechanical systems

AbstractAn approach to minimize the control costs and ensuring a stable deviation control is the Riccati controller and we want to use it to control constrained dynamical systems (differential algebraic equations of Index 3). To describe their discrete dynamics, a constrained variational integrators [1] is used. Using a discrete version of the Lagrange‐d’Alembert principle yields a forced constrained discrete Euler‐Lagrange equation in a position‐momentum form that depends on the current and future time steps [2]. The desired optimal trajectory (qopt, popt) and according control input uopt is determined solving the discrete mechanics and optimal control (DMOC) algorithm [3] based on the variational integrator. Then, during time stepping of the perturbed system, the discrete Riccati equation yields the optimal deviation control input uR. Adding uopt and uR to the discrete Euler‐Lagrange equation causes a structure preserving trajectory as both DMOC and Riccati equations are based on the same variational integrator. Furthermore, coordinate transformations are implemented (minimal, redundant and nullspace) enabling the choice of different coordinates in the feedback loop and in the optimal control problem. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Enhancing Digital Human Motion Planning of Assembly Tasks Through Dynamics and Optimal Control

Better operator ergonomics in assembly plants reduce work related injuries, improve quality, productivity and reduce cost. In this paper we investigate the importance of modeling dynamics when planning for manual assembly operations. We propose modeling the dynamical human motion planning problem using the Discrete Mechanics and Optimal Control (DMOC) method, which makes it possible to optimize with respect to very general objectives. First, two industrial cases are simulated using a quasi-static inverse kinematics solver, demonstrating problems where this approach is sufficient. Then, the DMOC-method is used to solve for optimal trajectories of a lifting operation with dynamics. The resulting trajectories are compared to a steady state solution along the same path, indicating the importance of using dynamics.

A Gait Optimization Smoothing Penalty Function Method for Bipedal Robot via DMOC .

For periodic gait optimization problem of bipedal walking robot, based on discrete mechanics and optimal control (DMOC), a class of smoothing penalty function method is proposed. The optimal control strategy and trajectory are solved by a new smoothing exact penalty function algorithm. The algorithm can quickly converge to a stable gait cycle independent the selection of the initial gait, otherwise, the algorithm only needs one step correction and then generate a stable gait cycle. Numerical simulation results show that the algorithm is feasible and effective. The algorithm makes the bipedal robot walk efficiently and stably on the even terrain.

Simultaneous path planning and trajectory optimization for robotic manipulators using discrete mechanics and optimal control

SUMMARYIn robotics, path planning and trajectory optimization are usually performed separately to optimize the path from the given starting point to the ending point in the presence of obstacles. In this paper, path planning and trajectory optimization for robotic manipulators are solved simultaneously by a newly developed methodology called Discrete Mechanics and Optimal Control (DMOC). In DMOC, the Lagrange–d'Alembert equation is discretized directly unlike the conventional variational optimization method in which first the Euler–Lagrange equations are derived and then discretization takes place. In this newly developed method, the constraints for optimization of a desired objective function are the forced discrete Euler–Lagrange equations. In this paper, DMOC is used for simultaneous path planning and trajectory optimization for robotic manipulators in the presence of known static obstacles. Two numerical examples, applied on a DELTA parallel robot, are discussed to show the applicability of this new methodology. The optimal results obtained from DMOC are compared with the other state-of-the-art techniques. The difficulties and problems associated in using the DMOC for Parallel Kinematic Machine (PKM) are also discussed in this paper.

An Improved DMOC Method with Gradient Penalty Term

An efficient method aimed at smooth and stable control forces for optimal control problem is described. Based on the native discrete mechanics and optimal control (DMOC) method, which focus mainly on the minimization of the total control forces, a gradient penalty term is introduced to cost functional to smooth the control forces. Then vibration of control forces is overcome by limiting the total gradient of the discrete control forces. With suitable discrete cost functional and constraints, the continuous optimal control problem is transformed to an equally finite dimensional form, which can be easily solved by standard algorithms. Finally, the numerical example of orbit transferring shows the effectiveness of the improved method.

Trajectory Design Combining Invariant Manifolds with Discrete Mechanics and Optimal Control

A mission design technique that combines invariant manifold techniques, discrete mechanics, and optimal control produces locally optimal low-energy trajectories. Previously, invariant manifolds of the planar circular restricted three-body problem have been used to design trajectories with relatively small midcourse change in velocity ΔV. A different method of using invariant manifolds is explored to design trajectories directly in the four-body problem. Then, using the local optimal control method DMOC (Discrete Mechanics and Optimal Control), it is possible to reduce the midcourse ΔV to zero. The influence of different boundary conditions on the optimal trajectory is also demonstrated. These methods are tested on a trajectory that begins in Earth orbit and ends in ballistic capture at the moon. Impulsive DMOC trajectories require up to 19% less ΔV than trajectories using a Hohmann transfer. When applied to low-thrust trajectories, DMOC produces an improvement of up to 59% in the mass fraction and 22% in travel time when compared with results from shooting methods.

Discrete Mechanics and Optimal Control and its Application to a Double Pendulum on a Cart

AbstractIn this paper we present a new approach to determine trajectories for changing the state of the double pendulum on a cart from one equilibrium to another and show the experimental realization on a test bench. The control of these transitions is accomplished by a two-degrees-of-freedom control scheme. For the design of the feedforward and feedback control of the system two models of the double pendulum on a cart are introduced. The feedforward control is achieved by the optimal control method Discrete Mechanics and Optimal Control (DMOC). The trajectories can be optimized with respect to energy consumption and transition time. Additionally, for the applicatory design, the implementation of a feedback control by means of gain-scheduling is explained. As experimental result the realization of a trajectory on the test bench is presented.

Computation of derivatives for structure preserving optimal control using automatic differentiation

AbstractIn this work we combine a recently developed method, Discrete Mechanics and Optimal Control (DMOC), with the well established Automatic Differentiation package ADOL‐C. DMOC is based on the discretization of the variational structure of the mechanical system which leads to structure (symplectic‐momentum) preserving time‐stepping equations. The discretized equations provide equality constraints for the resulting finite dimensional nonlinear optimization problem. For the solution of this problem standard nonlinear optimization techniques like sequential quadratic programming (SQP) are used. To ensure robustness of these techniques the computation of derivatives of the objective function and the constraints is of great importance. The concept of Automatic Differentiation (AD) is used to improve the performance of the SQP method. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Discrete mechanics and optimal control: An analysis

The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated.