TRAJECTORY PLANNING
5.7 HISTORICAL PERSPECTIVE
Example 5.10 Figure 5.19 shows the same six second trajectory as in Exam-ple 5.9 with the added constraints that the accelerations should be zero at the blend times.
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5.7 HISTORICAL PERSPECTIVE
The earliest work on robot planning was done in the late sixties and early sev-enties in a few University-based Artificial Intelligence (AI) labs [25, 28, 57].
This research dealt with high level planning using symbolic reasoning that was much in vogue at the time in the AI community. Geometry was not often explicitly considered in early robot planners, in part because it was not clear how to represent geometric constraints in a computationally plausible manner.
The configuration space and its application to path planning were introduced in [47]. This was the first rigorous, formal treatment of the geometric path
plan-HISTORICAL PERSPECTIVE 185 ning problem, and it initiated a surge in path planning research. The earliest work in geometric path planning developed methods to construct volumetric representations of the free configuration space. These included exact methods (e.g., [65]), and approximate methods (e.g., [11, 36, 47]). In the former case, the best known algorithms have exponential complexity and require exact de-scriptions of both the robot and its environment, while in the latter case, the size of the representation of C-space grows exponentially in the dimension of the C-space. The best known algorithm for the path planning problem, giving an upper bound on the amount of computation time required to solve the prob-lem, appeared in [12]. That real robots rarely have an exact description of the environment, and a drive for faster planning systems led to the development of potential fields approaches [39, 40].
By the early nineties, a great deal of research had been done on the geometric path planning problem, and this work is nicely summarized in the textbook [42]. This textbook helped to generate a renewed interest in the path planning problem, and it provided a common framework in which to analyze and express path planning algorithms. Soon after, the research field of Algorithmic Robotics was born at a small workshop in San Francisco [31].
In the early nineties, randomization was introduced in the robot planning community [5], originally to circumvent the problems with local minima in potential fields). Early randomized motion planners proved e↵ective for a large range of problems, but sometimes required extensive computation time for some robots in certain environments [38]. This limitation, together with the idea that a robot will operate in the same environment for a long period of time led to the development of the probabilistic roadmap planners [37, 58, 38].
Finally, much work has been done in the area of collision detection in recent years. [46, 52, 73, 74]. This work is primarily focused on finding efficient, incremental methods for detecting collisions between objects when one or both are moving. A number of public domain collision detection software packages are currently available on the internet.
Problems
MOTION PLANNING PROBLEMS TO BE WRITTEN
5-1 Show by direct calculation that the determinant of the coefficient matrix in Equation (5.43) is (tf t0)4.
5-2 Use Gaussian elimination to reduce the system (5.43) to upper triangular form and verify that the solution is indeed given by Equation (5.82).
5-3 Suppose we wish a manipulator to start from an initial configuration at time t0 and track a conveyor. Discuss the steps needed in planning a suitable trajectory for this problem.
5-4 Suppose we desire a joint space trajectory ˙qdi(t) for the i-th joint (assumed to be revolute) that begins at rest at position q0 at time t0 and reaches position q1 in 2 seconds with a final velocity of 1 radian/sec. Compute a cubic polynomial satisfying these constraints. Sketch the trajectory as a function of time.
5-5 Compute a LSPB trajectory to satisfy the same requirements as in Prob-lem 5-4. Sketch the resulting position, velocity, and acceleration profiles.
5-6 Fill in the details of the computation of the LSPB trajectory. In other words compute the portion of the trajectory between times tf tb and tf
and hence verify Equations (5.74).
5-7 Write a Matlab m-file, lspb.m, to generate an LSPB trajectory, given ap-propriate initial data.
5-8 Rewrite the Matlab m-files, cubic.m, quintic.m, and lspb.m to turn them into Matlab functions. Document them appropriately.
DYNAMICS 6
T
his chapter deals with the dynamics of robot manipulators. Whereas the kinematic equations describe the motion of the robot without consideration of the forces and torques producing the motion, the dynamic equations explicitly describe the relationship between force and motion. The equations of motion are important to consider in the design of robots, in simulation and animation of robot motion, and in the design of control algorithms. We introduce the so-called Euler-Lagrange equations, which describe the evolution of a me-chanical system subject to holonomic constraints (this term is defined later on). To motivate the Euler-Lagrange approach we begin with a simple deriva-tion of these equaderiva-tions from Newton’s Second Law for a one-degree-of-freedom system. We then derive the Euler-Lagrange equations from the principle of virtual work in the general case.In order to determine the Euler-Lagrange equations in a specific situation, one has to form the Lagrangian of the system, which is the di↵erence between the kinetic energy and the potential energy; we show how to do this in sev-eral commonly encountered situations. We then derive the dynamic equations of several example robotic manipulators, including a two-link cartesian robot, a two-link planar robot, and a two-link robot with remotely driven joints.
The Euler-Lagrange equations have several very important properties that can be exploited to design and analyze feedback control algorithms. Among these are explicit bounds on the inertia matrix, linearity in the inertia param-eters, and the so-called skew symmetry and passivity properties. We discuss these properties in Section 6.5.
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the dynamical equations of a robot, known as the Newton-Euler formulation which is a recursive formulation of the dynamic equations that is often used for numerical calculation.