Convex Optimization approach for Time-Optimal Path Tracking of Robots with Speed Dependent Constraints

Ardeshiri, Tohid; Norrlöf, Mikael; Löfberg, Johan; Hansson, Anders

doi:10.3182/20110828-6-it-1002.01136

Cited by 35 publications

(48 citation statements)

References 17 publications

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“…For this reason velocity limitsq have to be chosen, causing an unexploited potential. In [4] torque constraints are approximated with affine functions to consider them within a convex optimization. In contrast to this, we use the constraints provided by data sheets directly within a DP approach [5].…”

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confidence: 99%

A Dynamic Programming Approach for Time‐Optimal Path Following of Robots Considering Speed Dependent Torque Constraints

Oberherber

Gattringer

Müller

2017

Proc Appl Math and Mech

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Time-optimal path following, i.e. moving a robot's end-effector optimally along a specified geometric path, is a very important and well discussed problem in robotics. Nevertheless, most of the existing approaches concerning this topic neglect the speed dependency of torque constraints. The present paper presents a method for taking such constraints into account within a dynamic programming approach. To this end, the problem is treated in parameter space. This allows for an optimal use of existing resources. Due to the demanding constraints, precise mathematical models of the robots are indispensable. A satisfying match between model and real system can usually be achieved by parameter identification. For this purpose, it is a common way to derive the equations of motion using nominal parameters (masses, position of center of gravity, inertia and friction parameters), rewrite the equations in terms of linearly independent base parameters, and determine them with the help of measurements. Nevertheless, a parametrization of the motor torques has to be introduced in order to be able to consider their constraints within the optimization. In contrast to this, we present a general toolchain, based on the Projection Equation that directly derives the base parameter representation and furthermore the coefficients of the parametrized equations of motion. A verification in terms of a numerical example for a six-axis industrial robot demonstrate why speed dependent torque constraints are preferable over constant torque constraints for time-optimal robot trajectories. With a subsequent QR-decomposition, linearly independent base parameters p B are determined, resulting in Q = Θ B p B . In doing so, Θ B = ΘF Θ contains the linearly independent columns of Θ that are selected with the binary selection matrix F Θ . Numerical values for p B are then identified using a least squares error minimization.Parametrization of the EoM: Since the path is defined as a function of s, the EoM have to be parametrized along this path. The goal is to derive coefficients a Q (s), b Q (s), c Q (s) and d Q (s) so that the generalized torques can be written asA common approach is the insertion of specific values for q,q,q and g into Q = Θ(q,q,q)p, see e.g. [3]. This procedure has some drawbacks especially for systems with many DoF, since the resulting terms are not always easy to simplify. Alternatively, the parametrization of the EoM can elegantly be performed analytically using the previous mentioned method. For this reasonq(s) andq(s) are substituted into Θ Rc , Θ Rv , Θ T M and Θ T , where a separation concerning z , z and √ z is done. For indicating corresponding values an additional index is introduced. Terms concerning z are indexed with a, terms concerning z with b, terms concerning √ z with d and residual terms with c. After selecting the linearly independent columns of the parametrized information matrix with F Θ ,the coefficients follow by separation toThe matrices that are necessary for their calculation are mainly known from dynamic m...

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confidence: 99%

A Dynamic Programming Approach for Time‐Optimal Path Following of Robots Considering Speed Dependent Torque Constraints

Oberherber

Gattringer

Müller

2017

Proc Appl Math and Mech

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“…As an alternative to direct approach, the motion planning problem is often decoupled [2,3]. First, a high-level geometry path planner determines a path for the robot considering geometric constraints and ignoring system dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…Direct approaches [1] or one step methods solve the entire problem in one step. In general it is a highly difficult task, since both the geometric constraints (including collision avoidance), and the timing along this geometric path (including dynamic limitation) have to be optimized.As an alternative to direct approach, the motion planning problem is often decoupled [2,3]. First, a high-level geometry path planner determines a path for the robot considering geometric constraints and ignoring system dynamics.…”

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confidence: 99%

Path Tracking Algorithms for Non-Convex Waiter Motion Problem

Nagy

Csorvási

Vajk

2018

Period. Polytech. Elec. Eng. Comp. Sci.

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Originally, motion planning was concerned with problems such as how to move an object from a start to a goal position without hitting anything. Later, it has extended with complications such as kinematics, dynamics, uncertainties, and also with some optimality purpose such as minimum-time, minimum-energy planning. The paper presents a time-optimal approach for robotic manipulators.A special area of motion planning is the waiter motion problem, in which a tablet is moved from one place to another as fastas possible, avoiding the slip of the object that is placed upon it. The presented method uses the direct transcription approach for the waiter problem, which means a optimization problem is formed in order to obtain a time-optimal control for the robot. Problem formulation is extended with a non-convex jerk constraints to avoid unwanted oscillations during the motion. The possible local and global solver approaches for the presented formulation are discussed, and the waiter motion problem is validated by real-life experimental results with a 6-DoF robotic arm.Keywords motion planning, minimum-time control, time-optimal control, convex optimisation, robot control 1 Introduction Time-optimal motion planning has been a topic of active research since the 1980. Minimum-time algorithms can maximize productivity, and reduce energy consumption of the robotic system. Direct approaches [1] or one step methods solve the entire problem in one step. In general it is a highly difficult task, since both the geometric constraints (including collision avoidance), and the timing along this geometric path (including dynamic limitation) have to be optimized.As an alternative to direct approach, the motion planning problem is often decoupled [2,3]. First, a high-level geometry path planner determines a path for the robot considering geometric constraints and ignoring system dynamics. In the next stage (path tracking), a velocity profile for a predefined path is generated, where all constraints of the robot are applied for the fixed path. Decoupled approach is preferred to direct approach for its lower computational time.In this paper, we will focus on the second stage of the decoupled motion planning approach. Since the desired path of the robot is already defined, a scalar path coordinate

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“…Many efficient approaches for the time minimum trajectory planning (TMTP) problem along a predefined geometric path have been proposed, including the phase plane analysis approaches [3][4][5][6], the Pontryagin maximum principle and shooting algorithms [7], the convex optimization [2,10], the path reshaping [8], and the direct search method [9]. The recent works [2,11] are significant in which general and efficient methods are proposed.…”

Section: Introductionmentioning

confidence: 99%

Tractable Algorithm for Robust Time-Optimal Trajectory Planning of Robotic Manipulators under Confined Torque

Zhang

Jun

et al. 2014

INT J COMPUT COMMUN

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Convex Optimization approach for Time-Optimal Path Tracking of Robots with Speed Dependent Constraints

Cited by 35 publications

References 17 publications

A Dynamic Programming Approach for Time‐Optimal Path Following of Robots Considering Speed Dependent Torque Constraints

A Dynamic Programming Approach for Time‐Optimal Path Following of Robots Considering Speed Dependent Torque Constraints

Path Tracking Algorithms for Non-Convex Waiter Motion Problem

Tractable Algorithm for Robust Time-Optimal Trajectory Planning of Robotic Manipulators under Confined Torque

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