Investigations on Dynamics and Control of a Rimless Wheel Based 2D Dynamics Walker using Pulsed Torque Actuation
Wheeled systems are energy efficient on prepared surfaces like roads and tracks. Legged systems are capable of traversing different terrains but can be lossy. At low speeds and on off-road surfaces, legged systems using dynamic walking can be energy efficient. Towards this objective, the dynamics of the walker needs to be modelled and controlled. In addition, the braking and ground impact losses need to be minimized. This thesis presents analysis and experiments on the dynamics and control of a rimless-spoked-wheel based mobile robot (Chatur ∗) that belongs to a category between wheeled and legged systems. This rolling rimless wheel is effectively a 2D dynamic walker that serves as a platform for investigating the dynamics and energetics of inverted pendulum walking with constant step angle. A pulsed actuation torque is proposed for the system resulting in four torque regimes defined by the ratio of energy losses to available actuator torque. Five physical constraints that impose fundamental limits on the choice of operating points of a generic inverted pendulum walker are expounded and a method for locating optimal operating points is discussed. Chatur’s hardware design is elaborated and a control topology is proposed for pulsed actuation of the dual brushless dc (BLDC) motor driven platform with wheel synchronization. Various actuator torque profiles can be used to achieve dynamic ‘walking’ in a hub-actuated rimless wheel. The proposed pulsed actuation torque gives rise to four torque regimes that achieve sustained walking and a fifth regime where the walker keeps slowing down with each step. The regimes can be identified based on the fraction of stance phase for which the actuator is energized. Theoretical analysis and experimental results are presented. A simple closed-form analytical solution, using hyperbolic functions, is proposed for the stance phase inverted pendulum dynamics considering planar motion. Ground impacts are assumed to cause abrupt drop in velocity. A constant braking torque that lumps together the effect of several loss phenomena is also considered. Based on whether the CoM is rising or falling and whether or not there is an actuating torque, a stance phase can have four types of sub-phases — actuated rise, unactuated rise, actuated fall, unactuated fall. These are concatenated in four different ways to form repeating cycles yielding the four regimes. The experimental set-up is a fixed step-angle walker constructed using two synchronized adjacent rimless wheels independently actuated at the hub. Varying the magnitude and duty ratio of the torque pulse, the four proposed regimes are experimentally shown. The mechanical power consumption and cost of transport are computed from measured motor currents for different average forward speeds. Videos of the walks are also taken. The space of operating points for an inverted pendulum based bipedal dynamic walker in terms of constraints and optimality is investigated. The operating point of the walker can be specified by the combination of initial mid-stance velocity (v0) and step angle (φm) chosen for a given walk. Not all operating points lead to a realizable steady-state gait. Using basic mechanics, a framework of physical constraints that limit the choice of operating points is proposed. The constraint lines thus obtained delimit the valid region of operation of the walker in the v0–φm plane. Within this allowable region, sub-regions that result in various regimes of walking are identified. A given average forward velocity vx,avg can be achieved by several combinations of v0 and φm. Only one of these combinations results in the minimum mechanical power consumption and can be considered the opti-mum operating point for the given vx,avg. A method is proposed for obtaining this optimal operating point based on tangency of the power and velocity contours. Putting together all such operating points for various vx,avg, a family of optimum operating points, called the optimal locus, is obtained. For the energy loss and internal energy models chosen, the optimal locus obtained has a largely constant step angle with increasing speed but tapers oﬀ at non-dimensional speeds close to unity. Thus, choosing the right step angle and keeping it fixed over a broad range of speeds could lead to an inverted pendulum walker that is close to optimal from a mechanical energy perspective. The complete hardware design for Chatur and the caveats associated with reliable performance of the mechanical and electrical subsystems are elaborated. In order to en-sure lateral stability, the system uses two contralateral wheels each driven by a separate BLDC hub motor. From a motor drive perspective, the mechanical load belongs to a unique class of dynamic loads whose reflected torque has a characteristic cyclic varia-tion that repeats several times within a mechanical revolution. The proposed control topology has two hierarchical levels, an inner loop for torque control of BLDC motor implemented using a standard proportional-integral controller, and an outer loop for torque reference generation that uses the information on the ground impact instants and the motor position feedback. Ground impacts of the spokes are detected by an accelerometer to initiate the application of torque. The torque pulse magnitude can be set internally or by a manual operator via radio control. The pulse duration is programmable and enables attainment of various torque regimes at diﬀerent steady state speeds. The wheels are synchronized so that corresponding spokes on both wheels move in unison. This is achieved by including a wheel synchronization loop that compensates for any lag between the wheels. Lag is detected based on number of sector changes in the hall-eﬀect position sensor data received from both motors. An improved BLDC motor drive is developed wherein non-commutating current feedback is used to reduce current spikes during sector transitions. Experimental waveforms for controller validation are shown.
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