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In order to avoid motor saturation in turning maneuvers, an iterative Lamé-trajectory planning scheme is proposed to generate a smooth curvature-bounded transition trajectory for a differential-driving wheeled mobile robot (DWMR) switching from one straight path to another. The scheme consists of Lamécurve blending, inverse-kinematics computation, peak-torque positioning and torque-saturation avoidance. Firstly, a Lamé-curve blending procedure based on affine transformations, is formulated to generate a smooth G2-continuous transition trajectory for connecting two straight paths. Secondly, the platform twist is calculated according to the curvature of the Lamé-curve trajectory, then transformed into the actuated-joint rates by means of the inverse-kinematics model. Thirdly, a peak-torque positioning technique is developed to estimate the peak torques of the driving wheels when the DWMR tracks the trajectory, by combining the computed-torque method and the inverse-dynamics model. Finally, an iterative r-step saturation-avoidance prediction planning strategy is devised to suppress the peak motor torques, by means of two torque limitation schemes via adjusting trajectory curvature and robot speed. The simulation results show that, compared with the conventional planning techniques for circular arcs, our trajectory planning scheme can generate a smooth saturation-free transition trajectory with feasible curvature and traveling speed. The scheme is significantly beneficial for trajectory tracking under finite actuation torque in turning maneuvers, thereby preventing any possible path deviation caused by insufficient torque.

Logarithmic spirals are isogonal trajectories of pencils of lines. From a series of geometric consequences, we pick out a few which are relevant for kinematics: When a logarithmic spiral rolls on a line, its asymptotic point traces a straight line. Hence, wheels with the shape of a logarithmic spiral can be used for a stair climbing robot. When involute spur gears are to be generated by virtue of the principle of Camus, the auxiliary pitch curves must be logarithmic spirals. Two congruent logarithmic spirals can roll on each other while their asymptotic points remain fixed. A composition of two such rollings gives a two-parametric motion which allows a second decomposition of this kind. Some of these properties hold similarly for the spherical counterparts, the spherical loxodromes. For example, when in spherical geometry a loxodrome rolls on a circle, both asymptotic points trace circular involutes. Therefore, spherical loxodromes are auxiliary pitch curves for involute bevel gearing. On the other hand, spherical loxodromes can also be seen as helical curves in the projective model of hyperbolic geometry, where the sphere serves as a Clifford surface. This paves the way for remarkable arrangements of loxodromes on a sphere, e.g., a 3-web.

Logarithmic spirals are isogonal trajectories of pencils of lines. From a series of geometric consequences, we pick out a few which are relevant for kinematics: When a logarithmic spiral rolls on a line, its asymptotic point traces a straight line. Hence, wheels with the shape of a logarithmic spiral can be used for a stair climbing robot. When involute spur gears are to be generated by virtue of the principle of Camus, the auxiliary pitch curves must be logarithmic spirals. Two congruent logarithmic spirals can roll on each other while their asymptotic points remain fixed. A composition of two such rollings gives a two-parametric motion which allows a second decomposition of this kind. Some of these properties hold similarly for the spherical counterparts, the spherical loxodromes. For example, when in spherical geometry a loxodrome rolls on a circle, both asymptotic points trace circular involutes. Therefore, spherical loxodromes are auxiliary pitch curves for involute bevel gearing. On the other hand, spherical loxodromes can also be seen as helical curves in the projective model of hyperbolic geometry, where the sphere serves as a Clifford surface. This paves the way for remarkable arrangements of loxodromes on a sphere, e.g., a 3-web.

Since many off-the-shelf motor drives are supplied with complete control capability in the current, velocity and position loop, the robot model in the navigation control architecture can be oriented either to kinematics, interfaced with the velocity loop, or to dynamics, with the motor-current loop. Moreover, no constraints are imposed by a caster on the mobility of differential-driving mobile robots. Hence, a reduced model, containing only the platform, is sufficient for navigation control based only on the robot kinematics. However, if the multibody system model is used for navigation control based on the robot dynamics, to cope with the demands of high-speed manoeuvres and/or heavy-load operations, then the caster kinematics, especially the knowledge of the steering angle, is required to calculate the inertia matrix and the terms of Coriolis and centrifugal forces. While this angle can be measured by means of dedicated encoders to be installed for casters, the computation technique based on the existing tachometers, already mounted on the motor shafts for the servo control of the two driving wheels, is proved to be sufficient. Both a thorough kinematics model and a multibody dynamics model, including the platform and all different wheels, are formulated here for differential-driving mobile robots. Computational methods based on velocity compatibility and rigid body twists are proposed to estimate the steering angle. Simulation results of the differential-driving mobile robot moving on a smooth trajectory show the feasibility of the steering-angle computational scheme, which obviates the need of installing caster encoders. Moreover, a performance comparison on system modelling is implemented via simulation, between the differential-driving mobile robot model with and without caster dynamics. This further validates the importance of the dynamic effects of casters on the whole system model. Therefore, the multibody modelling approach for casters with the steering-angle computation technique can facilitate the navigation control architecture under dynamics conditions.

Applying computer vision to mobile robot navigation has been studied for over two decades. One of the most challenging problems for a vision-based mobile robot involves accurately and stably tracking a guide path in the robot limited field of view under high-speed manoeuvres. Pure pursuit controllers are a prevalent class of path tracking algorithms for mobile robots, while their performance is rather limited to relatively low speeds. In order to cope with the demands of high-speed manoeuvres, a multi-loop receding-horizon control framework, including path tracking, robot control, and drive control, is proposed in this paper. This is done within the vision guidance of differential-driving wheeled mobile robots (DWMRs). Lamé curves are used to synthesize a trajectory with G 2 -continuity in the field of view of the mobile robot for path tracking, from its current posture towards the guide path. The platform twist—point velocity and angular velocity—is calculated according to the curvature of the Lamé-curve trajectory, then transformed into actuated joint rates by means of the inverse-kinematics model; finally, the motor torques needed by the driving wheels are obtained based on the inverse-dynamics model. The whole multi-loop control process, initiated from Lamé-curve blending to computational torque control, is conducted iteratively by means of receding-horizon guidance to robustly drive the mobile robot manoeuvring close to the guide path. The results of numerical simulation show the effectiveness of our approach.

This paper focuses on the optimization of the McGill Schönflies Motion Generator. Recent trends on optimum design of parallel robots led us to investigate the advantages and disadvantages derived from an optimization based on performance indices. Particularly, we optimize here two different indices: the kinematic conditioning and the inertial conditioning, pertaining to the condition number of the Jacobian matrix and to that of the generalized inertia matrix of the robot, respectively. The problem of finding the characteristic length for the robot is first investigated by means of a constrained optimization problem; then plots of the kinetostatic and the inertial conditioning indices are provided for a particular trajectory to be tracked by the moving platform of the SMG. Deep connections appear between the two indices, reflecting a correlation between kinematics and dynamics.