Trajectory optimization

Depending on the configuration, open-chain robotic manipulators require a degree of trajectory optimization. For instance, a robotic arm with 7 joints and 7 links (7-DOF) is a redundant system where one cartesian position of an end-effector can correspond to an infinite number of joint angle positions, thus this redundancy can be used to optimize a trajectory to, for example, avoid any obstacles in the workspace or minimize the torque in the joints.[5]

Trajectory optimization is often used to compute trajectories for quadrotor helicopters. These applications typically used highly specialized algorithms.[6][7] One interesting application shown by the U.Penn GRASP Lab is computing a trajectory that allows a quadrotor to fly through a hoop as it is thrown. Another, this time by the ETH Zurich Flying Machine Arena, involves two quadrotors tossing a pole back and forth between them, with it balanced like an inverted pendulum. The problem of computing minimum-energy trajectories for a quadcopter, has also been recently studied.[8]

Trajectory optimization is used in manufacturing, particularly for controlling chemical processes (such as in [9] ) or computing the desired path for robotic manipulators (such as in [10] ).

There are a variety of different applications for trajectory optimization within the field of walking robotics. For example, one paper used trajectory optimization of bipedal gaits on a simple model to show that walking is energetically favorable for moving at a low speed and running is energetically favorable for moving at a high speed. [11] Like in many other applications, trajectory optimization can be used to compute a nominal trajectory, around which a stabilizing controller is built. [12] Trajectory optimization can be applied in detailed motion planning complex humanoid robots, such as Atlas. [13] Finally, trajectory optimization can be used for path-planning of robots with complicated dynamics constraints, using reduced complexity models. [14]

For tactical missiles, the flight profiles are determined by the thrust and lift histories. These histories can be controlled by a number of means including such techniques as using an angle of attack command history or an altitude/downrange schedule that the missile must follow. Each combination of missile design factors, desired missile performance, and system constraints results in a new set of optimal control parameters.[15]

Single shooting is the simplest type of trajectory optimization technique. The basic idea is similar to how you would aim a cannon: pick a set of parameters for the trajectory, simulate the entire thing, and then check to see if you hit the target. The entire trajectory is represented as a single segment, with a single constraint, known as a defect constraint, requiring that the final state of the simulation matches the desired final state of the system. Single shooting is effective for problems that are either simple or have an extremely good initialization. Both the indirect and direct formulation tend to have difficulties otherwise.[16][19] [20]

Multiple shooting is a simple extension to single shooting that renders it far more effective. Rather than representing the entire trajectory as a single simulation (segment), the algorithm breaks the trajectory into many shorter segments, and a defect constraint is added between each. The result is large sparse non-linear program, which tends to be easier to solve than the small dense programs produced by single shooting.[19][20]

Direct collocation methods work by approximating the state and control trajectories using polynomial splines. These methods are sometimes referred to as direct transcription. Trapezoidal collocation is a commonly used low-order direct collocation method. The dynamics, path objective, and control are all represented using linear splines, and the dynamics are satisfied using trapezoidal quadrature. Hermite-Simpson Collocation is a common medium-order direct collocation method. The state is represented by a cubic-Hermite spline, and the dynamics are satisfied using Simpson quadrature.[16][20]

Orthogonal collocation is technically a subset of direct collocation, but the implementation details are so different that it can reasonably be considered its own set of methods. Orthogonal collocation differs from direct collocation in that it typically uses high-order splines, and each segment of the trajectory might be represented by a spline of a different order. The name comes from the use of orthogonal polynomials in the state and control splines.[20][21]

In pseudospectral discretization the entire trajectory is represented by a collection of basis functions in the time domain (independent variable). The basis functions need not be polynomials. Pseudospectral discretization is also known as spectral collocation.[22][23][24] When used to solve a trajectory optimization problem whose solution is smooth, a pseudospectral method will achieve spectral (exponential) convergence.[25] If the trajectory is not smooth, the convergence is still very fast, faster than Runge-Kutta methods.[26][27]

In 1990 Dewey H. Hodges and Robert R. Bless [28] proposed a weak Hamiltonian finite element method for optimal control problems. The idea was to derive a weak variational form of first order necessary conditions for optimality, discretise the time domain in finite intervals and use a simple zero order polynomial representation of states, controls and adjoints over each interval.

Differential dynamic programming, is a bit different than the other techniques described here. In particular, it does not cleanly separate the transcription and the optimization. Instead, it does a sequence of iterative forward and backward passes along the trajectory. Each forward pass satisfies the system dynamics, and each backward pass satisfies the optimality conditions for control. Eventually, this iteration converges to a trajectory that is both feasible and optimal.[29]

When solving a trajectory optimization problem with an indirect method, you must explicitly construct the adjoint equations and their gradients. This is often difficult to do, but it gives an excellent accuracy metric for the solution. Direct methods are much easier to set up and solve, but do not have a built-in accuracy metric.[16] As a result, direct methods are more widely used, especially in non-critical applications. Indirect methods still have a place in specialized applications, particularly aerospace, where accuracy is critical.

One place where indirect methods have particular difficulty is on problems with path inequality constraints. These problems tend to have solutions for which the constraint is partially active. When constructing the adjoint equations for an indirect method, the user must explicitly write down when the constraint is active in the solution, which is difficult to know a priori. One solution is to use a direct method to compute an initial guess, which is then used to construct a multi-phase problem where the constraint is prescribed. The resulting problem can then be solved accurately using an indirect method.[16]

Single shooting methods are best used for problems where the control is very simple (or there is an extremely good initial guess). For example, a satellite mission planning problem where the only control is the magnitude and direction of an initial impulse from the engines.[19]

Multiple shooting tends to be good for problems with relatively simple control, but complicated dynamics. Although path constraints can be used, they make the resulting nonlinear program relatively difficult to solve.

Direct collocation methods are good for problems where the accuracy of the control and the state are similar. These methods tend to be less accurate than others (due to their low-order), but are particularly robust for problems with difficult path constraints.

Orthogonal collocation methods are best for obtaining high-accuracy solutions to problems where the accuracy of the control trajectory is important. Some implementations have trouble with path constraints. These methods are particularly good when the solution is smooth.