DESKFZG561 - theinvestmentmania.com … · “Robotics and Control”, Mittal R. K. & Nagrath I. J, TMH 2. “Fundamentals of Robotics: Analysis and Control”, Robert J., Schilling,

DESKFZG561 MECHANISMS & ROBOTICSKinematics of Manipulators–Lecture 14

dynamics of mechanism s and trajectory planning

Prof Milind RamgirWILP BITS PILANI , PUNE CENTER1

“Books :

1. “Robotics and Control”, Mittal R. K. & Nagrath I. J, TMH2. “Fundamentals of Robotics: Analysis and Control”, Robert J., Schilling, Prentice Hall

Solved problemS


Q.1 Calculate the velocity of the tip of the two-link, planar, RR manipulator arm

Solution – Assign frames, identify joint link parameters.Frame {2} is attached to the end of the manipulator. Length of each link is assumed 1, L1=L2=1

Solved problemS


Solved problemS - Since both the joints are rotary linear velocity is computed as


Solved problemS


Solved problemS


Solved problemS


Solved problemS


Solved problemS



Dynamic moDelingWhile Kinematics deals with finding position, velocity & acceleration based on geometrical constraints, dynamics is concerned with solving for these when an external force acts on the system or the system is released to evolve from some initial position(e.g. Pendulum).The dynamic behavior of the manipulator is acceleration, movement at constant speed and deceleration with time varying position and orientation of the manipulator.

The internal forces are caused by motion ( Velocity and acceleration) of the links. e.g.Inertial, Coriolis and frictional forces. And the external forces are caused dueenvironment/work piece. E.g. Load, Gravitational forces.

The time varying torque is applied at the joints to balance out the internal and externalforces. So that the links have to withstand stresses caused by force/torque balance acrossthese.The mathematical equations or manipulator dynamics are set of equations of motion is usefulfor computation of torque and forces required for execution of a typical work cycle, whichrequire for the design of links, joints drives and actuators.

The manipulator control maintains the dynamic response of the manipulator to obtain thedesired performance, which depends on the accuracy of the dynamic model and efficiency ofthe control algorithms.


Dynamic moDeling approachesAssumptions- Rigid body motion, No backlash, No friction, Effect of controlcomponent dynamics neglected, the resulting equations of motion are set of secondorder, coupled, nonlinear differential equations, consisting of inertia loading andcoupling reaction forces between joints.

Dynamics of rigiD boDy


Lagrangian mechanism


•This Method is based on energy.•Equations are obtained without considering the internal reaction forces.•It is ideal for more complex robotic manipulator configurations. e.g. Complex 3D robot, flexible link robot .•It is better than Newton's method for robotic applications.•It is based on differentiation of energy terms with respect to the systems variables & time. In this method we have to form the Lagrangian of the system, which is the difference of kinetic & potential energy of the system.

L = K.E.- P.E. = K - PL= Lagrangian , K.E.= Kinetic Energy, P.E.= Potential Energy.

The Lagrangian Euler dynamic formulation is based on set of generalizedcoordinates, Displacement q as joint variable which describes linear displacementd for prismatic joint, angular displacement θ for rotary joint, q’ describes linearvelocity d’(=v) and angular velocity θ’(=ω) for prismatic and rotary joints.Similarly generalized torque τ required at the joint to produce desired dynamicsrepresents the force f for prismatic joint and torque τ for revolute joint

Lagrangian mechanism


Since the Kinetic Energy and Potential Energies are the function of qi and q’i ( i- 1,2,3….n) so is the Lagrangian L.

The dynamic model based on Lagrange – Euler formulation is obtained from the Lagrangian, as set of equations,

The left hand side is sum of the torques/forces due to kinetic and potential energypresent in the system. The right hand side τi is the joint torque for joint I that isprovided by the actuator i.

If τi is zero, it means that joint I does not move and if τi ≠ 0, the manipulatormovement is modified by the actuator at joint i.

Lagrange - euler formulation


The Langrange – Euler formulation is a systematic procedure for obtaining thedynamic model of an n DOF manipulator. The n DOF open kinematic chain seriallink manipulator has n joint position or displacement variables, q = [ q1, …qn]T.

The LE formulation establishes the relation between the joint positions, velocities, accelerations and the generalized torques applied to manipulator.

The generalized torques are the nonconservative torques contributed by jointactuators , joint friction forces and induced joint torques. The induced joint torquesare the torques at the joint due to contact or interaction of the end effector with theenvironment.

The derivation of EOM using LE formulation is carried out using the linktransformation matrices T, which are obtained from the kinematic modeling. Firstthe link velocity is computed and next the link inertia tensor is obtained. These areused to compute K.E. Then P.E is calculated and next Lagrangian is formed, whichis substituted in LE eqn. to get the dynamic model.

Velocity of a point on the manipulator


Consider a n DOF manipulator with link i, Point P on the link. The frames are assigned Frame{0},{i-1} and {i}. The position vector iri describes the point P on the link with respect to frame {i}, in homogeneous coordinate notation,



We know in i-1Ti , qi = θi for Rotary joint and qi = di for prismatic joint. The velocity of point P w.r.t base coordinates, frame {0} is, 0vi, ,where ir’i = 0 is used.

The simplification steps of computation of partial derivative of the homogeneous transformation matrix.

Eqn. 6.26 can be obtained by •Interchanging row 1 with row 2,•Changing the sign of row1,•Making row3 and row4 zero.



Eqn. 6.26 can also be obtained by using matrix operations, using computers. Mathematically using 4X4 matrix Qj for revolute and prismatic joints

The result is same as eqn.6.26 hence



Similarly for Prismatic joint, Qj will be

The inerTia Tensor


The mass of the link contributes inertia forces during motion of the link. The inertial loads with respect to rotation about the origin of frame, are represented by a moment of inertia tensor. It is 4X4 matrix, which characterizes the distribution of mass of a rigid body (link i) The moment of inertia tensor Ii is

The kineTic energy


The kineTic energy



The poTenTial energy


The equaTion of moTion



Equation τi is the dynamic model of the manipulator and gives a set of non linear, coupled, second order ordinary differential equations for n links of the n DOF manipulator.These equations are the equations of motion or the dynamic equations of motions for the manipulator. Due to mass of the payload contributing to the inertia and its position is continuously changing is neglected in EOM.








The le dynamic model algoriThm


TrajecTory planning

INTRODUCTION

Path and trajectory planning means the way that a robot is moved from one location to another in a controlled manner, manipulator to follow a preplanned root.

The sequence of movements for a controlled movement between motion segment, in straight-line motion or in sequential motions.

It requires the use of both kinematics and dynamics of robots.

The goal of trajectory planning is to describe the requisite motion of the manipulator as a time sequence of joint/link/end effector locations and derivatives of locations, which are generated by interpolating or approximating the desired path by polynomial function.


TrajecTory planning

The user inputs a list of parameters and constraints describing the desired trajectory. It is expected to generate a time sequence of intermediate configurations expressed in either joint or Cartesian coordinate frames.


TrajecTory planning Path: A sequence of robot configurations in a particular order without regard to the timing of these configurations. Also it is locus of points (either in joint space or in the Cartesian space) to be traversed by the manipulator to execute the specific task. A path is a purely geometric (spatial) description of the motion.

Trajectory: It concerned about when each part of the path must be attained,thus specifying timing. A trajectory is the time sequence of position, velocity andacceleration for each joint or end effector of the manipulator. Trajectory is bothspatial and temporal representation of motion. It can be specified either in jointspace or in Cartesian space.

Knot Points or Via Points: The set of intermediate locations between thestart and goal points on the trajectory through which the manipulator must passenroute to the destination is also called as interpolation points.

Spline: It is the smooth function that passes through the set of via points.

Joint Space Trajectory Planning: Each point is specified in terms of adesired position and orientation of the end effector frame relative to the baseframe. Each point is converted into a set of a desired joint positions by applicationof inverse kinematics. Smooth function is then found for each of the joints, whichpasses through these points.


TrajecTory planning

Cartesian Space Trajectory Planning: The path is explicitly specifiedin the Cartesian Space. The path constraints ( Velocity, acceleration etc.) arespecified in Cartesian coordinates and the joint actuators are servoed in jointcoordinates to the specified trajectory.

Trajectory Generation : It is the act of computing the trajectory as a timesequence of values in real time, using the trajectory planning algorithm based onthe spatial and temporal constraints.

Path Update Rate : The rate at which the trajectory points are computed atrun time.

StepS in trajectory planning

(i) Task Description- a) Motion identification, Point to point motion(PTP), Continuous path motion (CP), Path Points (Initial, Goal and Via Points) , b) travel time description.

(ii) Selecting and Employing a trajectory Planning Technique- Joint Space for PTP or Cartesian Space Techniques for CP.

(iii) Computing the trajectory. Compute the time sequence of values attained by the function generated from the trajectory planning techniques.


TrajecTory planning

JOINT-SPACE VS. CARTESIAN-SPACE DESCRIPTIONSJoint-space description: - The description of the motion to be made by the robot by its joint values.- The motion between the two points is unpredictable. Cartesian space description: - The motion between the two points is known at all times and controllable.- It is easy to visualize the trajectory, but is difficult to ensure that singularity.- Provides the time history of the location, velocity and acceleration of the end effector with respect to base.

Sequential motions of a robot to follow a straight line.

Cartesian-space trajectory (a) The trajectory specified in Cartesian coordinates may force the robot to run into itself, and (b) the trajectory may requires a sudden change in the joint angles.


TrajecTory planning JOINT-SPACE TECHNIQUESTo plan the trajectory, the value of joint variables have to be determined from the end effector location specified by the user.

Step I- Obtain the corresponding set of joint variable values for each specified path point, by inverse kinematics algorithm or recording of teaching by showing technique.

Step II- Find the smooth function q(t) for each of the joints of an n DOF manipulator, which interpolates the joint variable vector for each given path point, with respect to imposed constraints. Features required in joint space trajectory planning algorithm

1. The travelling time between any two path points is same for each joint so as to make all the joints reach their corresponding path joint location simultaneously.

2. Jlint Locations and velocities be contineous functions of time ( continuity of accelerations may be inposed, too) so as to ensure smooth motion.

3. The interpolating function should not be computationally intensive.4. Non smooth trajectories and other similar undesirable effects be minimized.


The end effector initial position is qs and it is moved to goal position qg in certain amount of time. There can be innumerable function that satisfy the given criteria.The selection of appropriate polynomial function has two common approachesa) Choose the highest degree polynomial with at least as many coefficients as

are the specified constraints. ( Polynomial more oscillatory and numerically less accurate)

b) Split the trajectory into segments and use lower degree polynomial to interpolate each segment with smooth transition from one segment to next segment. ( Imposes additional constraint required for blending two segments for smooth motion)

TrajecTory planning JOINT-SPACE TECHNIQUES

1) Cubic polynomial- cubicsconnect the path points of each joint in smooth way,

2) A linear function with parabolic blends- each segment between two successive path points contains a linear function with parabolic blends near the path points.


In Cartesian space Techniques the position and orientation of a rigid body can be clearly defined.

In the planning to enable the manipulator’s end effector to track a given path is investigated.

The user specifies the desired end effector path, the travelling time, and the tool orientations along the path.

To tackle the above problem, a parametric description for the desired end effector path is required.

This description specifies the spatial attributes of the requisite motion with respect to the base frame.

First the mathematics of parametric description of path in Cartesian Space is reviewed e.g. Straight line path, Circular path,

Then the common techniques for trajectory planning is carried out like Position planning, Orientation Planning.

TrajecTory planning- CARTESIAN SPACE TECHNIQUES


Joint space versus cartesian space TrajecTory planning

Joint Space Trajectory1. Simple to implement and joint

coordinates fully specify the position and orientation of the end effector.

2. Does not account for the existence of obstacles in the workspace of a manipulator.

3. Due to the non linear nature of the manipulator’s kinematics model, it is difficult to predict the resulting end effector motion that will be produced by a particular trajectory executed in the joint space.

Cartesian Space Trajectory1. Easy to specify the position and

orientation of a rigid body in space, and better for specifying the a task or identifying an obstacle in work space.

2. The obstacles and path followed by the end effector can be expressed in simple transformations and easy to visualize.

3. Computation is complex. Manipulator control system controls joint variables, need to revert to inverse kinematics /dynamics, reduces the sampling rate and trajectory tracking accuracy.


Joint space versus cartesian space TrajecTory planning

Joint Space Trajectory Cartesian Space Trajectory4. The planned paths may have

singularities or may pass close to manipulator singularities. When singular configuration of manipulator is reached, one or more joint velocities may increase infinitely.

5. Requires more complex algorithms and more computational ability, so that in run time the joints are servoed to follow the desired Cartesian paths. It offers a better and simpler representation of the task but at higher computational cost.

Documents

DESKFZG561 - theinvestmentmania.com … · “Robotics and Control”, Mittal R. K. & Nagrath I. J, TMH 2. “Fundamentals of Robotics: Analysis and Control”, Robert J., Schilling,