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Mechanical Engineering |
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Lagrangian Dynamics
Lagrangian dynamics is a mathematical problem solving method used to analyze the dynamics of moving objects (treated as particles) or multi-body dynamic systems (also commonly known as mechanisms). Unlike the Newtonian method, the Lagrangian method can be used to solve multi-body dynamics problems without illustrating the free-body diagram and calculating all the joint forces. Instead, such a method is used to solve dynamics problems by considering the initial and final kinetic and potential energies of moving objects at the center of mass. For example, the Lagrangian method can be used to calculate the required input torques from the motors of a robot arm. The Lagrangian method extends beyond the kinetic and potential energy analysis for multi-body dynamic systems. Lagrangian dynamics is commonly used to design robotic systems, such as (but not limited to) robotic arms. Introduction to Robotics is an example of a fourth-year mechanical and aerospace engineering elective that teaches Lagrangian dynamics.
Lagrangian dynamics of a dual-arm robot
Figure 1 depicts the configuration of a dual-arm robot with a known position of the end effector (indicated as point P) in both x and y directions relative to the fixed pin-joint, linkage lengths, center of mass positions, mass and the rotational moment of inertia of the arms. Additionally, the angular velocities and accelerations of the linkages are also known, where the velocities and accelerations are the actuator (the arm motors in this case) inputs. Furthermore, the orientations of the arms are known, where the orientations are evaluated using inverse kinematics. Lastly, out-of-plane motion is not considered. Lagrangian dynamics is to be used to calculate the required torques for both the lower and upper arms by following the steps outlined below:
< Figure 1: Configuration of the dual-arm robot. Joint B is a kinematic joint. >
Step 1: State the equations of motion at the center of mass for both the lower and upper arms.
Lower Arm Motion
Upper Arm Motion
where,
Step 2: Differentiate the equations of motion with respect to time to obtain the velocity equations at the center of mass for both of the arms.
Consider the lower arm
The lower arm orientation is the only parameter that is time-dependent.
Consider the upper arm
Both the lower and upper arm orientations are time-dependent parameters. All the other parameters are independent of time.
Step 3: Derive and simplify the expressions for kinetic and potential energies at the center of mass for both of the robot arms. Consider the lower arm
The kinetic energy at the center of mass is expressed as follows:
Back-substituting equations 9 and 10 into equation 13 and implying that sin(x)^2 + cos(x)^2 = 1,
The potential energy at the center of mass is expressed as follows when point A is treated as the reference point:
Back-substituting equation 4 into equation 15,
Consider the upper arm
The kinetic energy at the center of mass is expressed as follows:
Back-substituting equations 11 and 12 into equation 17 and implying that sin2x + cos2x = 1, cos(x+y) = cos(x) cos(y) - sin(x) sin(y), and cos(x) = cos(-x),
The potential energy at the center of mass is expressed as follows when point A is treated as the reference point:
Back-substituting equation 8 into equation 19,
Step 4: Back-substitute the kinetic and potential energy equations into the Lagrangian equation. The Lagrangian equation is the difference between the sum of kinetic and potential energies of the arms at the center of mass, as shown below:
Derivative of equation 24 with respect to time
Derivative of the Lagrangian expression with respect to the lower arm orientation
Derivative of the Lagrangian expression with respect to the angular velocity of the upper arm
Derivative of equation 27 with respect to time
Derivative of the Lagrangian expression with respect to the orientation of the upper arm
Step 6: Calculate the required torque for both the lower and upper arms using the following equations: The required torque for actuation of the lower arm
The required torque for actuation of the upper arm
Reference :
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