Simple Trajectory Generation

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The world is full of PID-loops, thermostats, and PLLs. These are all feedback loops where we control a certain output variable through an input variable, with a more or less known physical process (sometimes called "plant") between input and output. The input or "set-point" is the desired output where we'd like the feedback system to keep our output variable.

Let's say we want to change the set-point. Now what's a reasonable way to do that? If our controller can act infinitely fast we could just jump to the new set-point and hope for the best. In real life the controller, plant, and feedback sensor all have limited bandwidth, and it's unreasonable to ask any feedback system to respond to a sudden change in set-point. We need a Trajectory - a smooth (more or less) time-series of set-point values that will take us from the current set-point to the new desired value.

Here are two simple trajectory planners. The first is called 1st order continuous, since the plot of position is continuous. But note that the velocity plot has sudden jumps, and the acceleration & jerk plots have sharp spikes. In a feedback system where acceleration or jerk corresponds to a physical variable with finite bandwidth this will not work well.

1st_order_trajectory

We get a smoother trajectory if we also limit the maximum allowable acceleration. This is a 2nd order trajectory since both position and velocity are continuous. The acceleration still has jumps, and the jerk plot shows (smaller) spikes as before.

2nd_order_trajectory

Here is the python code that generates these plots:

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# AW 2014-03-17
# GPLv2+ license
 
import math
import matplotlib.pyplot as plt
import numpy
 
# first order trajectory. bounded velocity.
class Trajectory1:
	def __init__(self, ts = 1.0, vmax = 1.2345):
		self.ts = ts		# sampling time
		self.vmax = vmax	# max velocity
		self.x = float(0)	# position
		self.target = 0
		self.v = 0 			# velocity
		self.t = 0			# time
		self.v_suggest = 0
 
	def setTarget(self, T):
		self.target = T
 
	def setX(self, x):
		self.x = x
 
	def run(self):
		self.t = self.t + self.ts	# advance time
		sig = numpy.sign( self.target - self.x ) # direction of move
		if sig > 0:
			if self.x + self.ts*self.vmax > self.target:
				# done with move
				self.x = self.target
				self.v = 0
				return False
			else:
				# move with max speed towards target
				self.v = self.vmax
				self.x = self.x + self.ts*self.v
				return True
		else:
			# negative direction move
			if self.x - self.ts*self.vmax < self.target:
				# done with move
				self.x = self.target
				self.v = 0
				return False
			else:
				# move with max speed towards target
				self.v = -self.vmax
				self.x = self.x + self.ts*self.v
				return True
 
	def zeropad(self):
		self.t = self.t + self.ts
 
	def prnt(self):
		print "%.3f\t%.3f\t%.3f\t%.3f" % (self.t, self.x, self.v )
 
	def __str__(self):
		return "1st order Trajectory."
 
# second order trajectory. bounded velocity and acceleration.
class Trajectory2:
	def __init__(self, ts = 1.0, vmax = 1.2345 ,amax = 3.4566):
		self.ts = ts
		self.vmax = vmax
		self.amax = amax
		self.x = float(0)
		self.target = 0
		self.v = 0 
		self.a = 0
		self.t = 0
		self.vn = 0 # next velocity
 
	def setTarget(self, T):
		self.target = T
 
	def setX(self, x):
		self.x = x
 
	def run(self):
		self.t = self.t + self.ts
		sig = numpy.sign( self.target - self.x ) # direction of move
 
		tm = 0.5*self.ts + math.sqrt( pow(self.ts,2)/4 - (self.ts*sig*self.v-2*sig*(self.target-self.x)) / self.amax )
		if tm >= self.ts:
			self.vn = sig*self.amax*(tm - self.ts)
			# constrain velocity
			if abs(self.vn) > self.vmax:
				self.vn = sig*self.vmax
		else:
			# done (almost!) with move
			self.a = float(0.0-sig*self.v)/float(self.ts)
			if not (abs(self.a) <= self.amax):
				# cannot decelerate directly to zero. this branch required due to rounding-error (?)
				self.a = numpy.sign(self.a)*self.amax
				self.vn = self.v + self.a*self.ts
				self.x = self.x + (self.vn+self.v)*0.5*self.ts
				self.v = self.vn
				assert( abs(self.a) <= self.amax )
				assert( abs(self.v) <= self.vmax )
				return True
			else:
				# end of move
				assert( abs(self.a) <= self.amax )
				self.v = self.vn
				self.x = self.target
				return False
 
		# constrain acceleration
		self.a = (self.vn-self.v)/self.ts
		if abs(self.a) > self.amax:
			self.a = numpy.sign(self.a)*self.amax
			self.vn = self.v + self.a*self.ts
 
		# update position
		#if sig > 0:
		self.x = self.x + (self.vn+self.v)*0.5*self.ts
		self.v = self.vn
		assert( abs(self.v) <= self.vmax )
		#else:
		#	self.x = self.x + (-vn+self.v)*0.5*self.ts
		#	self.v = -vn
		return True
 
	def zeropad(self):
		self.t = self.t + self.ts
 
	def prnt(self):
		print "%.3f\t%.3f\t%.3f\t%.3f" % (self.t, self.x, self.v, self.a )
 
	def __str__(self):
		return "2nd order Trajectory."
 
vmax = 3 # max velocity
amax = 2 # max acceleration
ts = 0.001 # sampling time
 
# uncomment one of these:
#traj = Trajectory1( ts, vmax )
traj = Trajectory2( ts, vmax, amax )
 
traj.setX(0) # current position
traj.setTarget(8) # target position
 
# resulting (time, position) trajectory stored here:
t=[]
x=[]
 
# add zero motion at start and end, just for nicer plots
Nzeropad = 200
for n in range(Nzeropad):
	traj.zeropad()
	t.append( traj.t )
	x.append( traj.x ) 
 
# generate the trajectory
while traj.run():
	t.append( traj.t )
	x.append( traj.x ) 
t.append( traj.t )
x.append( traj.x )
 
for n in range(Nzeropad):
	traj.zeropad()
	t.append( traj.t )
	x.append( traj.x ) 
 
 
# plot position, velocity, acceleration, jerk
plt.figure()
plt.subplot(4,1,1)
plt.title( traj )
plt.plot( t , x , 'r')
plt.ylabel('Position, x')
plt.ylim((-2,1.1*traj.target))
 
plt.subplot(4,1,2)
plt.plot( t[:-1] , [d/ts for d in numpy.diff(x)] , 'g')
plt.plot( t , len(t)*[vmax] , 'g--')
plt.plot( t , len(t)*[-vmax] , 'g--')
plt.ylabel('Velocity, v')
plt.ylim((-1.3*vmax,1.3*vmax))
 
plt.subplot(4,1,3)
plt.plot( t[:-2] , [d/pow(ts,2) for d in numpy.diff( numpy.diff(x) ) ] , 'b')
plt.plot( t , len(t)*[amax] , 'b--')
plt.plot( t , len(t)*[-amax] , 'b--')
plt.ylabel('Acceleration, a')
plt.ylim((-1.3*amax,1.3*amax))
 
plt.subplot(4,1,4)
plt.plot( t[:-3] , [d/pow(ts,3) for d in numpy.diff( numpy.diff( numpy.diff(x) )) ] , 'm')
plt.ylabel('Jerk, j')
plt.xlabel('Time')
 
plt.show()

See also:

I'd like to extend this example, so if anyone has simple math+code for a third-order or fourth-order trajectory planner available, please publish it and comment below!

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