BENG 221 Grant Vousden-DishingtonProblem Solving ReportThe Minimum-Jerk Trajectory for n-DOF Reaching Movements via Calculus of VariationsPROBLEM STATEMENTEfficient and accurate motor control remains an elusive solution within the field of robotics, and the success of a control system depends heavily on what metric is used to define the path taken by the motor plant. Calculating the desired movements of a robot with three or more degrees of freedom is an extremely complex problem, but the central nervous system controls dozens of degrees of freedom very quickly. As such, there is much interest in discovering the policy used by the motor nervous system and how it changes. Because fields, including bioengineering, aim to understand, implement, and even restore such functions in disabled patients, validating computational models of trajectory planning policy in simulation is paramount to that discovery.In the motor control framework, this policy must include a description of the trajectory the end effector must take – here, “end effector” refers to the (zero-)point on the arm that we desire to reach the target destination. By default, the end effector is the hand or gripper of the system, but it may just as well be the functional point of a tool, like a hammer or wrench, or another point on the limb, such as the center of the forearm or shins. The trajectory of a system is more complex than simply naming the Cartesian coordinates along the path; a trajectory function must describe when the end effector is expected to be at these positions. The problem here is that, given any starting and ending position in Cartesian space, there are infinitely many trajectories. Namely, we can make the trajectory as smooth or noisy as we like, even when there are some constraints on where we can move to.This is the motivation for formulating the trajectory-forming problem as an optimization problem. Though not necessarily unique, solutions for this type of problem will at least meet the criteria good enough for our purposes. Thus, the next part of the problem is to decide what must be optimized. In motor control, there are primarily two outputs that can be optimized: jerk and energy. Optimizing jerk entails finding the smoothest curve connecting an initial position, velocity, and acceleration to a desired position, velocity, and acceleration over an already specified time interval. Minimum energy models are similar, but they are more complex in that the energy expended by a plant is dependent on the architecture of the actuators/musculature, and the time-constraints for these models is more relaxed, since shorter time intervals typically involve expending more energy because a higher peak velocity is needed. Of key importance is that minimum-energy models necessarily rely on the dynamics of the system but minimum-jerk models can suffice on a simply kinematic description. It is for this reason that I use a simplified minimum-jerk model instead.CALCULUS OF VARIATIONS THEORY AND SETUPCalculus of variations is similar to gradient descent methods in differential equation analysis. Like gradient descent, it is a first order approach focused on minimizing the gradient of a given function∇ F. However, in the case of gradient descent, F may be any type of function, but calculus of variations deals specifically with functionals F, maps which take other functions as parameters and return elements that form the basis of that function's vector space, which is typically a scalar, as it will be in this formulation. The functional will serve as a 'cost' function that needs to be minimized. For this experiment, the cost function being minimized is jerk, which is defined as the third-derivative of position, the rate of change of acceleration.x (t )We define jerk as only a function of time to simplify the calculations. In realistic conditions, other factors will determine the jerk, such as forces due the the environment and muscle weakness and fatigue. As such, the definition of jerk is easily extendable to other domains and partial differential equations. In order to turn this definition of jerk into a cost, we must ensure that it is always positive and take the totality of this measurement over the entire movement. As each movement is continuous, we define this sum as an integral over the time of the movement tI to tf, the initial and final time points, respectively.C (t)=∫titf(x (t))2dtCalculus of variations works by finding not the minimum of this functional, but its minimum with respect to an arbitrary perturbation function, which we will call u(t). This perturbation function is multiplied by a perturbation constant α and added to the original jerk function. Below is shown the total function's new form, as well as a graph of how it affects the overall position and trajectory.The plot, modified from Shadmehr et al. (2005), shows the original functionx (t )in black, the pertubation function u(t) in pink, and the sum of the two, with an α equal to 1 (green) and greater than 1 (orange).x (t)+αu (t)Thus, the final form of the functional that we will use for this problem has the form C (r (t))→C (r (t)+α u (t ))To be useful, we must first apply some constraints to the perturbation function. The key aspects we wish to capture, as shown in the plot above, are that the perturbation function and all of its defined derivatives vanish at the endpoints ti and tf. Because we are dealing with the third-derivative of position, we need to make sure u(t) and its first three derivatives are zero.u (ti)=˙u(ti)=¨u (ti)=u (ti)=0u (tf)=˙u(tf)=¨u(tf)=u (tf)=0 Just as in single-variable optimization and gradient descent, calculus of variations takes the derivative of the cost functional. Instead of just seeking where the functional itself equals 0, however, we also evaluate the integral when the perturbation constant α equals 0, and use integration by parts to find which derivative of position must equal zero for the cost (i.e. jerk) to be minimized. In addition to this mapping from function to functional, a critical step to using calculus of variations that allows usto determine which derivative of x(t) must equal zero is the fundamental lemma of calculus of variations. This lemma states the following: If it is the case that, given a k-times continuously differentiable function f,∫abf (x )h( x)dx=0 for every k-times continuously differentiable function h(x) such that h(a) = h(b) = 0, then f(x)