A proof of the structure of the minimum-time control law of robotic manipulators using a Hamiltonian formulation

A Brief Introduction to Variational Integrators

“…2.3 and 3.5 for a precise definition of continuous and discrete symplecticity. Examples of Hamiltonian systems appearing in science and engineering include but are not limited to the structural biology (Gay-Balmaz et al 2009), molecular dynamics (Stavros 2014;Manning and Maddocks 1999), mathematical models in ecosystem dynamics (Kirwan 2008), superconductivity (Bogolyubov 1972), plasma physics (Larsson 1996), celestial mechanics and cosmology (Arnold et al 2006), fluid mechanics (Desbrun et al 2014;Gawlik et al 2011), mechanics of materials and structures , theoretical physics (Esposito et al 2004;Marsden 1988), aerospace engineering (Kasdin et al 2005), satellite dynamics and control (Kuang et al 2003;Koon et al 2011), kinematics and dynamics of mechanisms and robots (Macchelli et al 2009;Chen 1990) and other areas of seismic (Luo et al 2013), mechanical and electrical (Clemente-Gallardo and Scherpen 2003) engineering. Symplectic integrators are methods specially formulated to produce a symplectic flows on the phase space.…”

Smooth time-optimal tool trajectory generation for CNC manufacturing systems

“…The acceleration constrained TOTG problems have been proven that the optimal control has "bang-bang" or "bang-singular-bang" structures (Chen and Desrochers [5], Bobrow et al [6]). Also, it has been proven that these discontinue control structures may generate vibrations and large contour errors in high-speed machine systems.…”

“…Because of the high demand of maximum productivity in the CNC machining, the time optimal trajectory generation (TOTG) process plays an important role in motion planning problem (Dong and Stori [2], Jamhour and André [3], and Smith et al [4]). The ordinary TOTG processes only consider the constraints which limit the acceleration values, whereas the time optimal acceleration strategies of these trajectories have been proven to be "bangbang" or "bang-singular-bang", which is not physically realizable (Chen and Desrochers [5], Bobrow et al [6]). Direct implement of a "bang-bang" trajectory on a physical system with non-specialized controller can induce tool vibrations and overshoot of the nominal acceleration limits.…”

Time Optimal Path-Tracking Control of Kinematically Redundant Manipulators

“…When the motion of the system is not at singular points and arcs (10), (18) , the time optimal solution to the transformed problem is bang-bang in the controlũ. That is, at any instant,ũ =s min orũ =s max .…”

“…When the motion of the system is not at singular points and arcs (10), (18) , the time optimal solution to (a) Time trajectoryṡ(s) (b) Concept of the obtainedθ r (θ r ) Fig. 1 Generation of the path-tracking time trajectory and concept of the consequentlyobtainedθ r (θ r ) for the case of one redundant degree of freedom the transformed problem must also be bang-bang in the controlũ.…”