International Journal of Control, Automation, and Systems (2013) 11(4):805-814 DOI 10.1007/s12555-011-0065-y ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555 Algebraic Solution to Minimum-Time Velocity Planning Gabriele Lini, Aurelio Piazzi*, and Luca Consolini Abstract: The paper poses the problem of minimum-time velocity planning subject to a jerk amplitude constraint and to arbitrary velocity/acceleration boundary conditions. This problem which is relevant in the field of autonomous robotic navigation and also for inertial one-dimensional mechatronics systems is dealt with an algebraic approach based on Pontryagin’s Maximum Principle. The exposed complete solution shows how this time-optimal planning can be reduced to the problem of determining the posi- tive real roots of a quartic equation. An algorithm that is suitable for real-time applications is then pre- sented. The paper includes detailed examples also highlighting the special cases of this planning prob- lem. Keywords: Autonomous robotic navigation, feedforward control, iterative steering, minimum-time control, Pontryagin’s Maximum Principle, velocity planning. 1. INTRODUCTION In the field of robotics, much research has focused on minimum-time problems because solving them helps to achieve high-performance or to maximize production. For instance, an effective algorithm to move a robotic manipulator in minimum-time along a specified geometric path subject to input torque/force constraints was presented by Shin and McKay in [1]. In another instance, taken from autonomous robotic navigation, a unicycle mobile robot subject to linear and angular velocity bounds can be moved in minimum-time from a given configuration to another one by using Reeds- Shepps curves [2,3]. Still in the field of autonomous robotic navigation, the work presented in this paper faces the minimum-time (longitudinal) velocity planning in the context of the so- called path-velocity decomposition [4] and the iterative steering technique [5-7]. The robot vehicle has to travel on an assigned geometric path of a given length and the longitudinal vehicle velocity can be planned by seeking a time-optimal (i.e., minimum-time) motion subject to a jerk amplitude constraint and to arbitrary velocity/ acceleration boundary conditions. The continuous- acceleration profile with a given bound on the jerk (i.e., the time derivative of the acceleration) facilitates the implementation of the time-optimal movement and the interpolation of arbitrarily given velocity and acceler- ation at the endpoints of the planned time interval permits the supervisor of the iterative steering strategy to perform real-time sensor-based navigation while ensuring an overall smoothness of the vehicle motion [5]. The importance of minimum-time velocity planning with arbitrary boundary conditions was first pointed out by Koh et al. [8] in 1999 in the context of robotic and mechatronics systems with real-time motion planning applications. In subsequent years, velocity planning for autonomous navigation was dealt with cubic (polyno- mial) splines schemes and parametric local optimization to achieve minimum-time optimality in static and dynamic environments [9] or minimum-jerk optimality when a prefixed time interval is given [10-12]. These approaches use fast and efficient optimization algorithms and can deal with velocity and acceleration constraints, but they all cannot achieve true optimality because the function space where to search the optimizer is restricted by the choice of the polynomial splines scheme and moreover, the optimization algorithms can only converge on local minima. An application of [10] to the velocity planning for autonomous passenger vehicles was presented in [13] to achieve travel comfort with low values of acceleration and jerk. For an automated assembly manufacturing, a fifth order splines scheme was adopted in [14] to obtain a velocity planning that minimizes the time integral of the squared jerk. A comprehensive reference on velocity planning for automatic machines and robots can be found in [15]; more general trajectory planning methods and algorithms are presented in [16]. The addressed minimum-time velocity planning problem will be easily recast into an input-constrained reachability control problem for the triple integrator system (cf. (6)), where the time-optimal control input is actually the second time-derivative (jerk) of the sought © ICROS, KIEE and Springer 2013 __________ Manuscript received July 1, 2011; revised July 25, 2012 and March 8, 2013; accepted March 11, 2013. Recommended by Edi- torial Board member Duk-Sun Shim under the direction of Editor Hyouk Ryeol Choi. This paper is a revised and expanded version of a work pre- sented at the Seventh International Workshop on Robot Motion and Control, Czerniejewo (Poland), 1-3 June 2009 [17]. Gabriele Lini is with SISSA, International School for Ad- vanced Studies, Via Bonomea 265, I-34136, Trieste, Italy (e-mail: gabriele.lini@sissa.it). Aurelio Piazzi and Luca Consolini are with the Department of Information Engineering, University of Parma, Parco Area delle Scienze 181A, I-43124, Parma, Italy (e-mails: {aurelio.piazzi, luca.consolini}@unipr.it). * Corresponding author.