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Aerial Robotics

Rigid Body Dynamics

C. Papachristos

Robotic Workers (RoboWork) Lab

University of Nevada, Reno

CS-791

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Rigid Body Dynamics

  • We need a mathematical model to describe the relationship between a robot’s driving forces (thrust, wind, external disturbances, …) and its motion evolution (translational & rotational velocities)

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  • Different systems have different methods of generating motion:
    • Propellers / Ducted Fans
    • Aerodynamic Surfaces
    • Actuated Joints

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  • Start with a simple 4-Static Force Vector Rigid Body (e.g. Quadrotor)

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Kinematics & Dynamics

  • Kinematics:
    • Provide motion of the robot without consideration of forces or torques producing the motion

Important to consider for animation, navigation control, guidance, robot design, etc.

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  • Dynamics:
    • Describe relationship between forces and motions

Important to consider for simulation, optimal control, robot design etc.

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Kinematics & Dynamics

Dynamic Model Formulations

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  • Newton-Euler:
    • Yields recursive algorithm to derive dynamics (e.g. for multi-joint robots), computationally simple

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  • Lagrange:
    • Conceptually straightforward, systematic development of model independent of reference frame

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Rigid Body

 

 

 

 

 

(simple Linear Motion)

Linear Motion

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Rigid Body

 

 

 

 

 

 

 

 

Rotational Motion

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Simplified Rigid Body Dynamics Example: MAV Robot

 

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Simplified Rigid Body Dynamics Example: MAV Robot

Unconstrained (floating-base) in the general case

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  • Underactuated System:
    • 6 Degrees-of-Freedom, 4 actuators to control them
    • Generally (regardless of number of actuators), impossible to drive 1 DoF independently of all the rest

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We leverage dynamics couplings

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  • Goal
    • Develop relationship between forces moments and rigid body motion for a particular robot configuration

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Simplified Rigid Body Dynamics Example: MAV Robot

  • Assumption 1:

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The robot is flying as a rigid body with negligible aerodynamic effects on it – for the operating airspeeds

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  • Assumption 2:

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The propeller is considered as a simple disc that generates thrust and a moment around its shaft

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  • Model:

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  • We have:

 

 

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Simplified Rigid Body Dynamics Example: MAV Robot

  • Remember: We already have Rigid Body Kinematic Equations:

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Simplified Rigid Body Dynamics Example: MAV Robot

 

 

 

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Simplified Rigid Body Dynamics Example: MAV Robot

 

 

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Simplified Rigid Body Dynamics Example: MAV Robot

 

 

 

Use�slide 12

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Simplified Rigid Body Dynamics Example: MAV Robot

 

 

 

 

 

 

 

 

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Simplified Rigid Body Dynamics Example: MAV Robot

 

 

Shorthand for the rule

 

 

 

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Simplified Rigid Body Dynamics Example: MAV Robot

 

 

 

 

 

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Simplified Rigid Body Dynamics Example: MAV Robot

 

 

Centrifugal

(acceleration)

Euler (acceleration)

Coriolis (acceleration)

 

 

 

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Simplified Rigid Body Dynamics Example: MAV Robot

 

 

 

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Simplified Rigid Body Dynamics Example: MAV Robot

 

 

 

 

 

 

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Simplified Rigid Body Dynamics Example: MAV Robot

 

 

 

 

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Simplified Rigid Body Dynamics Example: MAV Robot

 

 

 

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Simplified Rigid Body Dynamics Example: MAV Robot

  • Newton-Euler Formulation

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Replacing in:

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We take:

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where:

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Simplified Rigid Body Dynamics Example: MAV Robot

 

 

 

 

 

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Newton-Euler Dynamics:

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Finally, we append Forces and Moments

Simplified Rigid Body Dynamics Example: MAV Robot

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Forces (in Body Frame):

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Moments (in Body Frame):

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Simplified Rigid Body Dynamics Example: MAV Robot

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Forces (in Body Frame):

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Moments (in Body Frame):

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Simplified Rigid Body Dynamics Example: MAV Robot

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Rigid Body Dynamics – Wrench

 

 

 

 

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Rigid Body Dynamics – Wrench

 

 

 

 

Remember:

 

 

 

 

“Spatial” Wrench

“Body” Wrench

(1)

Substitute�in (1):

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Rigid Body Dynamics – Wrench

  • Examples:

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Rigid Body Dynamics – Newton Euler

 

 

 

 

 

 

 

 

and

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Rigid Body Dynamics – Newton Euler

 

 

 

 

Simplified Inertia Matrix (approx.�with principal moments of inertia):

 

 

(by equating Kinetic�Energy in 2 frames)

 

 

Spatial Inertia Matrix

    • Symmetric
    • Positive-Definite

 

  • Note: Rigid Body Kinetic Energy expressed w.r.t. Twist & Spatial Inertia:

Spatial Momentum of Rigid Body

 

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Lie Bracket

 

 

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Lie Bracket

 

 

 

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Lie Bracket

 

 

 

 

 

corresponds to:

 

 

 

corresponds to:

Where:

 

for any:

 

 

 

 

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Rigid Body Dynamics – Newton Euler

Newton-Euler Formulation

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  • Rigid Body Wrench in linear form:

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  • Can also be expressed in terms of:

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  • Change of coordinate frame (equate Kinetic Energy)

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Spatial Inertia Matrix

 

 

 

 

Spatial Inertia Matrix expressed in {S} frame

 

: Dynamics Equation has same form regardless of coordinate frame

: “Inverse Dynamics” for Rigid Body

 

: “Forward Dynamics” for Rigid Body

 

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Time for Questions !

CS-791

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