Three most important problems in dextrous manipulation by multifingered robotic hands are the optimal grasp synthesis problem, real-time grasping force optimization problem, and coordinated manipulation with finger gaiting.

(a) Optimal grasp synthesis problem - Given an arbitrary object and the corresponding friction models, plan a set of contacts for the hand such that the grasp is force closure if such grasp exists. Moreover it is optimal in some sense. Traditional heuristic approaches, although being successfully applied to several classical manipulation examples, encounter difficulties in systems with more than 2 fingers and complex object geometries. In Chapter 3, we introduce several candidate grasp quality functions and formulate the grasp synthesis problem as a Max-transfer, a Max-normal-grasping-force, and a Min-analytic-center problem. The physical meaning of each quality function is explained. Each problem assumes the form of max - min - max or min - max - min type. Then, we develop algorithms for computing the gradients of these quality functions. When real-time solutions are needed for applications such as contact points servoing in coordinated manipulation [51, 50], we introduce two simplified quality functions, along with several examples. Note that the optimal solutions of the simplified problems coincide with previous results obtained using heuristic approaches, demonstrating again generality of our current methods. Finally, we perform experimental studies on the HKUST three-fingered hand using real-time optimization of the simplified quality functions.

(b) Real-time grasping force optimization problem - Given a set of contact points of the hand, the external wrench, and the corresponding friction models, determine an optimal grasping force in real-time with respect to a given objective function. In Chapter 4, we formulate the grasping force optimization as a convex optimization problem on the Riemannian manifold of positive definite matrices subject to linear constraints, for which five algorithms are developed and their respective step size selection are discussed in detail. However, the needs for a valid initial condition are found to be a common problem for all these five algorithms. We then propose two methods to the initial condition problem. Convergence analysis are performed to show the quadratic convergence of some of the five algorithms and exponential convergence of their continuous versions. Finally, simulation and experiments are conducted to compare the computation time, convergence rates, and accuracy of these five algorithms.

(c) Coordinated manipulation with finger gaiting - Given a periodic motion of the object, the geometry of the object and fingers, and the workspaces of fingers of the hand, determine if finger gaits are necessary. If that, how to plan a sequence of discrete events (fingers make or break contacts with the object) as well as the trajectories of fingers and contacts between any two events such that (1) all finger trajectories are contained in their respective workspaces; (2) at any time, whatever number of fingers are in contact with objects, the respective grasp keeps force closure. In Chapter 5, we give some preliminary results on this problem. We first use stratified configuration spaces to model the manipulation system with finger gaiting. Then, for a given gait and a periodic motion of the object, we construct two behavior basis for each finger of the hand corresponding to the contact and non-contact states. The state equation, and the trajectory of each finger are globally the sum of the two components: that when this finger is in contact with the object and that when it is away from the object. By doing so, we translate the gait feasibility problem into the feasibility of a set of nonlinear constraints. In Chapter 5, we also give several possible future works on the optimal grasp synthesis and planning of finger gaiting.

By solving these three problems and combining all related algorithms, multi-fingered robotic hands can be controlled to fulfill manipulation in a more dextrous and efficient way.

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