Non-holonomic Kinematics and the Role of Elliptic Functions in Constructive Controllability

A Mathematical Introduction to Robotic Manipulation

“…For the case of m = 2, Brockett and Dai [14] have shown that the optimal inputs are elliptic functions (see also the next section). However, we can extend Algorithm 1 to this case as follows: 2.…”

Autonomous Parafoil Guidance in High Winds

“…Much of the inspiration and key insight into the rich history of the problem are found in the work of Jurdjevic [17], which includes a more general geometric version of the necessary condition derivation in Chapter 4. The work of Brockett [7] details the role of elliptic integrals and functions in deriving closed form solutions to optimal control problems of this kind.…”

Planning Motions for Robotic Systems Subject to Differential Constraints

“…we can use a nonlinear static state feedback in order to further simplify (8). Define the control input τ as…”

“…More in general, this is another example of the minimalistic trend in the field of robotics, aimed at designing devices of reduced complexity for performing complex tasks. The archetype of rolling manipulation is the plate-ball system [31,27,24,8]: the ball (the manipulated object) can be brought to any contact configuration by maneuvering the upper plate (the first finger), while the lower plate (the second finger) is fixed. Despite its mechanical simplicity, the planning and control problems for this device already raise challenging theoretical issues.…”