Robot Dynamics
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by Bharath Kumar P. Last updated on 23/Oct/2021
Posted on 23/Oct/2021
Robot dynamics is concerned with the relationship between the forces acting on a robot mechanism and the accelerations they produce. Typically, the robot mechanism is modeled as a rigid-body system, in which case robot dynamics is the application of rigid-body dynamics to robots. The two main problems in robot dynamics are:
Forward dynamics is also known as "direct dynamics," or sometimes simply as "dynamics." It is mainly used for simulation. Inverse dynamics has various uses, including online control of robot motions and forces, trajectory design and optimization, design of robot mechanisms, and as a component in some forward-dynamics algorithms.
Dynamics of a Robot Car:
Image Credits: therobotreport
Given that the dynamic properties of a mobile robot have an important influence on its motion, mainly from the wheel engines traction forces FR, FL, it is possible to incorporate them into the mobile robot’s dynamic model. According to the second Newton's law
Image Credits: humusoft
The linear acceleration of the mobile robot’s base can be interpreted as a derivation of its linear velocity v and interpreted as a sum.
Image Credits: humusoft
while the traction forces FR, FL can be defined from the robot’s overall torque
Image Credits: humusoft
where τR, τL express partial wheel torques. Those torques can be defined in terms of the traction forces as
Image Credits: humusoft
while the left engine traction force FL is considered in opposite direction to suit the relation for the robot’s overall torque τrobot,
Image Credits: humusoft
Angular acceleration of mobile robot ω ? can be expressed by its torque τrobot and robot’s overall moment of inertia J as
Image Credits: humusoft
By substituting the partial wheel torques for right and left engine τR, τL into the sum of total mobile robot’s torque τrobot and by comparing it with the equation, it is possible to obtain the angular velocity ω ? of a two-wheeled mobile robot as a function of a robot’s engines traction forces.
Image Credits: humusoft
Mathematical model of mobile robot with differentially driven wheels, that consist of a kinematic and dynamic, the model may be represented by block diagram
Image Credits: humusoft
Wheel Friction:
Image Credits: researchgate
Friction acting against the movement is dependent on the object’s mass by the normal force FN and it can be classified as static or kinetic friction. Static friction is considered only at zero velocity and it is represented by its coefficient μs as a force threshold Fs, which must be overcome to set the object, in this case, a mobile robot, moving. Until reaching this force threshold, the effect of wheel traction force Fext is compensated by static friction force Ffr. After overcoming the Fs, the robot is set in motion, now affected by kinetic friction, which is usually represented as constant Coulomb friction, defined as
Image Credits: humusoft
where μk is the Coulomb friction coefficient. This kinetic friction model defines the friction force at non-zero velocities, it is independent of the size of the contact area and always acts against the movement, however, the friction force Ffr is dependent on a moving object’s velocity direction, no matter the size. In general, the coefficients are defined in terms μs ≥ μk, while in the case of μs > μk it is possible to consider the effect of stiction,
Image Credits: humusoft
The kinetic model of friction can be further extended by considering the viscous friction, defined by μv coefficient, which causes an increase in the friction force with increasing velocity v of moving object. Finally, at small velocities and with the active stiction effect, it is possible to apply the Stribeck curve defined by coefficient ast, which replaces the step change of the friction force from Fs to Fk by step-less change at a small velocity interval. By combining the above-mentioned friction models, a generalized model of friction can be obtained as
Image Credits: humusoft
and it can be represented as a function of velocity v.
Image Credits: humusoft
Image Credits: roboticgizmos
Drone Dynamics:
Image Credits: researchgate
Quadcopter basically holds a rigid cross-linked structure that has four independent rotors with fixed pitched propellers. Among the four propellers, two are rotating in a clockwise direction while the other two rotate in the anti-clockwise direction. The control of the quadcopter is obtained by changing the angular speed of the propellers Ω?? (?? = 1, 2, 3, 4). The rotational movement of the quadcopter along X, Y, and Z axes can be described by roll (??), pitch (??), and yaw angle (??).
Image Credits: iopscience
Every controller input has an effect on a certain movement such as ??2 effects on roll movement, ??3 effects on pitch movement, ??4 effects on yaw movement and ??1 has an effect on upward movement along Z-axis. Here, as the ‘+’ (plus) configuration is chosen for this quadcopter, the control inputs produce the effects on the system.
Image Credits: iopscience
The mathematical model of the quadcopter is,
Image Credits: iopscience
Another kinematic relationship is required between Euler rates, [??, ??, ??]T on the earth fixed frame and angular velocity, [??, ??, ??]T of the quadcopter to describe the whole complete system.
Image Credits: iopscience
The complete dynamic model of the quadcopter can be described by four control inputs, ?? = [??1??2??3??4]?? and 12 state vectors, ???? = [?? ?? ?? ??? ??? ??? ?? ?? ?? ?? ?? ??]??.
Image Credits: pinterest
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