This document discusses trajectory generation methods for servo motor drives. It describes requirements like smooth motion between points A and B within speed, acceleration and jerk limits. It proposes a method using symmetric jerk profiles of 8 zones to generate trajectories in real-time while accounting for issues like overshoot, small spans and variable inertia. Experimental results show the method generating trajectories of up to 30 revolutions while meeting requirements for velocity, position and varying inertia.

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Trajectory generation for Servo motor
drives

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Requirement
• To move (linear or rotary )motor from point A to B
• Generate a trajectory (periodic position command )
• Smooth motion
• As fast as possible
• Limit jerks, acceleration and speed to specified value
• Load inertia is known but variable

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Requirements.
• Minimum distance A to B can be small
• Real time computation (though motor is running)
• Long trajectories are possible (limited processor memory real
time while motor is moving calculations)
• Trajectories are always from standstill to standstill
• Trajectory is defined as instantaneous value of position, velocity
and acceleration to be commanded to the control system

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• Floating point (Pseudo (IQ) or true) support
• Digital signal processor (finite periodic sampling time ~ 1ms)
• Memory is limited to 500 samples
• Trajectory generation and data use occurs in a circular buffer
• Trajectory generation will fill buffer (provided it is empty ) at 10 X
data consumption
Given

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Divide trajectory into regimes
e.g. for positive direction movement
• Constant positive jerk (acceleration increase )
• Constant positive acceleration (vel increase ,jerk =0)
• Constant negative jerk (acceleration decrease to 0)
• Constant positive velocity(cruise mode) (jerk=0,acc=0)
• midpoint
Method

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• Constant positive velocity (cruise mode) (jerk=0,acc=0)
• Constant negative jerk (acceleration decrease to –ve value)
• Constant negative acceleration (vel decrease ,jerk = 0 )
• Constant positive jerk (acceleration increase to 0)
• Stop is position B
Method.

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Ideal case
Zone 1 2 3 4 5 6 7 8
Zone1: Standby
Zone2: Const +ive jerk
Zone3: Const +ive acc
Zone4: Const -ive jerk
Zone5: Const vel
Zone6: Const -ive jerk
Zone7: Const -ive acc
Zone8: Const +ive jerk
Jerk
velocity
position
acceleration

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Simple calculations using basic equations of motion
a(k) = a(k-1) + J(k-1).T
v(k) = v(k-1) +a(k-1).T
S(k) =S(k-1) + v(k-1).T + 1/2 . a(k-1) . T . T
Where
J=jerk ,a = acc, v =velocity ,S = position
Mirror image based solution
Real time computation (with motor running)
Limiting to maximum permissible limits for (j, a, v)
Advantages

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• Overshoot /undershoot at the end (numerical errors , finite
sampling, slight unsymmetry between the first and second half) do
not allow velocity and acceleration to converge to 0 simultaneously
• Small Span trajectories cannot be supported (as midpoint can be
hit before cruise mode)
• Variable inertia -> maximum acceleration varies  hence span
varies
• Jerky trajectory termination
Challenges

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Zone 1 2 3 4 5 6 7 8 9 10
Undershoot due to numerical errors
Zone1: Init
Zone2: Const +ive jerk
Zone3: Const +ive acc
Zone4: Const -ive jerk
Zone5: Const vel
Zone6: Const -ive jerk
Zone7: Const -ive acc
Zone8: Const +ive jerk
Jerk
velocity
position
acceleration
A
B
midpoint

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Jerk
acceleration
velocity
position
B
A
Zone 1 2 3 4 5 6 7 8 9 10
Overshoot due to numerical errors
Zone1: Init
Zone2: Const +ive jerk
Zone3: Const +ive acc
Zone4: Const -ive jerk
Zone5: Const vel
Zone6: Const -ive jerk
Zone7: Const -ive acc
Zone8: Const +ive jerk
midpoint

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• Real time compensation for over/undershoot for smooth
termination
• Smooth correction
• Errors due to discrepancies in symmetry fixed in real time
Proposed method

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• Smaller spans supported (by backward scaling)
• Variable inertia programmable (by scaling)
• Trajectory is almost always nearly symmetric
• Simple computation
• Fixed and floating point implementations
Proposed method

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Minimum span
Zone 1 2 3 4 5 6 7 8
Zone1: Init
Zone2: Const +ive jerk
Zone3: Const +ive acc
Zone4: Const -ive jerk
Zone5: Const vel
Zone6: Const -ive jerk
Zone7: Const -ive acc
Zone8: Const +ive jerk
midpoint

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• Servo motor drives
• Machine tool industry
• Position control
• Robotics
Target Application

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Experimental Results
• 30 revolutions
• 3X inertia
• forward direction
• 3200 rpm cruise
speed
Red : actual velocity
Blue: actual position

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Experimental Results.
• 30 revolutions
• 3X inertia
• 3200 rpm cruise speed
• opposite direction
Red : actual velocity
Blue: actual position

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Experimental Results..
Red : velocity
Blue: position
• 5 revolutions
• 3X inertia
• 3200 rpm cruise speed
• span near limiting span
Red : actual velocity
Blue: actual position

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Experimental Results…
• 5 revolutions
• 3X inertia
• 3200 rpm cruise speed
• near limiting span
• opposite direction
Red : actual velocity
Blue: actual position

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Thank You
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