QUIZ What does Jacobian matrix represent What does
QUIZ What does Jacobian matrix represent? What does singularities represent? How to determine singularities of manipulator?
STATIC FORCES & TORQUES
STATIC FORCES & TORQUES When a robot perform a given task, such as lifting up a workpiece from a machine, its end-effector exerts moment and force to the external environment at the point of contact. The force and moment are generated by the actuators installed at the various joints. In statics, the relationship between the JOINT torques/forces, and the CARTESIAN moments and the forces at the end-effector is sought. It serves as a basis for sizing the links and bearings of the manipulator and for selecting appropriate actuators. Introduction to Robotics-SK SAHA-Tata Mc. Graw-Hill
STATIC FORCES & TORQUES fi : force applied at link i by link i-1 fi+1 : force applied at link i+1 by link i
STATIC FORCES & TORQUES Newton third law: Action & Reaction Principle -i+1 fi+1 : force applied at link i by link i+1 Forces acting on the link i, The different between in xi+1 yi+1 zi+1 and is that is expressed in xiyizi. is expressed
STATIC FORCES & TORQUES Moment exerted on link i by the link i-1 Moment exerted on link i+1 by the link i Moments acting on link i,
STATIC FORCES & TORQUES Moments acting on link i, When link i experiencing the reaction force –fi+1 , this force may turn the link i about the joint axis zi+1. Therefore the term is the torque (moment of force), which reflects the turning effect of the reaction force, .
STATIC FORCES & TORQUES Forces acting on the link i, Moments acting on link i, Given the force vector fi and the moment ni acting on the link i, the force fi+1 and the moment ni+1 acting on the link i+1 can be directly computed.
STATIC FORCES & TORQUES- Example 5. 2 -1, Man Zhihong Link αi ai di ϴi 1 α 1 0 d 1 ϴ 1 2 α 2 0 d 2 ϴ 2 3 α 3 0 d 3 ϴ 3 The joint distances at the current structure are: d 1=0. 5 m, d 2=-0. 5 m, d 3=1 m Assume ϴ 1=ϴ 2=0
STATIC FORCES & TORQUES Translation vectors, The force acting on link 2 is, The moment acting on link 2 is,
STATIC FORCES & TORQUES The force acting on link 1 is, The moment acting on link 1 is, The force acting on the robot base is,
STATIC FORCES & TORQUES The moment acting on the robot base is, The 3 rd components of M 0, M 1 and F 2 should be resisted by the three joints. The 1 st and 2 nd components are resisted by the structures of the robot arm itself. 1 st and 2 nd joint are rotary joints, the third is the prismatic joint. In order to keep the robot arm in static equilibrium, the torques and force produced by the joint actuators should be as follow:
SUMMARY When robot interact with environment, external forces and moments may act on the end-effector. Forward kinematics and static equilibrium, estimate the force and torque transmitted from the robot hand down to the robot base. Finally, we calculate the joint torques and/or forces, to keep the robot manipulator in static equilibrium.
RELATIONSHIP BETWEEN FORCE, TORQUE AND JACOBIAN IN STATIC We have understood that Jacobian matrix relates the velocities (linear & angular velocities) expressed in Cartesian Coordinate system and joint velocities (angular velocity for revolute joint & linear velocity for prismatic joint) expressed in Joint Coordinate system. JACOBIAN
RELATIONSHIP BETWEEN FORCE, TORQUE AND JACOBIAN IN STATIC In some books, the Jacobian matrix could also be used to represent the relationship between end effector infinitesimal displacement (linear and rotational) in Cartesian and infinitesimal displacement in Joint Coordinate system. 6 x 1 vector nx 1 vector
RELATIONSHIP BETWEEN FORCE, TORQUE AND JACOBIAN IN STATIC Lets, Work is the product of vector force and vector displacement, thus,
RELATIONSHIP BETWEEN FORCE, TORQUE AND JACOBIAN IN STATIC Work is the product of vector force and vector displacement, thus, We also know, Finally,
QUIZ Point C, S, and P are the reference points of the camera, hand tool and the product, respectively. The object as seen by the camera is represented by matrix . The robot base as seen by the camera is represented by matrix, Determine the transformation of P with respect to point B? .
ROBOT DYNAMICS Concern with the relationship about forces acting on robot mechanism and accelerations they produced. Normally involved two: FORWARD DYNAMICS INVERSE DYNAMICS Given the forces, workout the acceleration Given acceleration, workout the forces
ROBOT DYNAMICS Develop the mathematical model to represent the movement of the manipulator. Established a set of equations of motion (EOM) that describe the dynamic response of the manipulator to input actuator torques. Manipulator’s EOM Langrange-Euler (Energy Based) Newton-Euler (Force Balance)
LANGRANGIAN EULER METHODS A scalar function called Langrange function or Langrangian L, L=K-P is the difference between the total kinetic energy , K and the total potential energy, P of a mechanical system.
LANGRANGIAN EULER METHODS The dynamic model based on Langrange-Euler formulation is obtained from the Langrangian, as a set of equations, Langrangian function L of a system of p particles in Cartesian space can be expressed as follows,
LANGRANGIAN EULER METHODS y A set of generalized coordinates q, ϴ x First order derivatives of x, y with respect to time, t.
LANGRANGIAN EULER METHODS
LANGRANGIAN EULER METHODS Approximate to linear equation
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