ERK'2019, Portorož, 70-73 70
Optimal velocity proﬁle planning considering velocity,
acceleration and jerk constraints
Martina Loknar, Gregor Klanˇ car, Saˇ so Blaˇ ziˇ c
Faculty of Electrical Engineering, Trˇ zaˇ ska cesta 25, SI-1000 Ljubljana, Slovenia
E-mail: martina.loknar@fe.uni-lj.si
Abstract
The aim of this study is to determine time-minimal veloc-
ity proﬁle for a wheeled mobile robot whose movement
complies with velocity, acceleration and jerk constraints
and is restricted to an arbitrary predeﬁned path. The pro-
posed two-step algorithm ﬁrst enables the determination
of the velocity proﬁle that respects the speed and accel-
eration constraints and in the second step additionally
applies the limitation of the jerk.
1 Introduction
Optimal control theory describes the application of forces
to a system for the purpose of maximizing some measure
of performance or minimizing a cost function [1]. It is
an extension of the calculus of variations and this math-
ematical optimization method is largely due to the work
of [2]. Using Pontryagin’s maximum principle, in [3] the
authors demonstrated that all time-optimal motions of the
mobile platform with two independently driven wheels
occur for controls that are at each instant on the upper or
lower limit.
Mechanical systems, including biological systems,
have maximum allowances related to various dynamic
variables, above which the components of the system may
begin to fail. Excess of jerk thresholds is associated with
mechanical wear, tool life repercussion and adverse ef-
fect on the actuator performance, degradation of machine
junctions, and in biological systems tear of ligaments and
muscles or breakage of bones [4], [5]. Minimizing jerk
is therefore beneﬁcial in reducing stress and wear of the
mechanical components, extending machine or tool life,
reducing position tracking errors, minimizing the excita-
tion of vibrations in general, enabling better quality ﬁn-
ishes in machining tasks and ensuring that the motor is
able to provide the requested current fast enough. Me-
chanical and robotics engineers have recognized the ben-
eﬁts of jerk minimization and therefore prefer to design
jerk-limited proﬁles [6].
Authors in [7] investigated time optimal two-stage
path planing under kinematic and dynamic constraints
and obtained the shortest path and a time optimal velocity
proﬁle. A path planning technique to minimise the time
of reaching the end point in desired direction and with de-
sired velocity is presented in [8]. In [9] the authors sug-
gested a methodology to generate minimum time optimal
velocity proﬁles for a vehicle with prescribed accelera-
tion limits along a speciﬁed path. Similarly, a method
for minimum-time velocity planning with velocity, ac-
celeration and jerk constraints was proposed, based on
a sequence of linear programming feasibility checks, de-
pending on motion constraints and generic boundary con-
ditions [10].
This paper outlines a new approach of determining
time-minimal optimal velocity proﬁle for a wheeled mo-
bile robot on a predeﬁned path. We believe that we have
found an innovative solution that is easy to implement
and computationally undemanding with coherent course
of calculation that relies heavily on analytical expressions
of given physical quantities. The ﬁrst step of the algo-
rithm ensures that the resulting velocity proﬁle complies
with speed and acceleration constraints. The additional
jerk constraints are considered in the second step of the
algorithm, where the acceleration discontinuities are elim-
inated by smoothing the velocity proﬁle from the ﬁrst
step.
2 Problem formulation
Let the motion of a particle along a three times continu-
ously differentiable plane curveC be described as a func-
tion of timet2 [0;t
f
] by the position vector r(t) mea-
sured from a given ﬁxed origin. Velocity v(t), acceler-
ation a(t) and jerk j(t) vectors can be expressed in the
tangential-normal form as:
v(t) =v(t)  ^
T (1a)
a(t) =a
T
(t)  ^
T+a
R
(t)  ^
N (1b)
j(t) =j
T
(t)  ^
T+j
R
(t)  ^
N; (1c)
where
^
T and
^
N are the unit tangent vector and the unit
normal vector, respectively:
^
T(t) =
v(t)
kv(t)k
;
^
N(t) =
_
^
T(t)
k
_
^
T(t)k
: (2)
Given a feasible path from initial to ﬁnal point, the opti-
mization problem is to ﬁnd the velocity proﬁlev(t) that
reaches the end of the path in minimum time in a way that
none of the velocity, acceleration or jerk constraints from
71
Eqs. (3a;3b;3c) are violated:
0   k v(t)k   v
MAX
; 8t2 [0;t
f
]; (3a)
a
2
T
(t)
a
2
T
MAX
+
a
2
R
(t)
a
2
R
MAX
  1; 8t2 [0;t
f
]; (3b)
j
2
T
(t)
j
2
T
MAX
+
j
2
R
(t)
j
2
R
MAX
  1; 8t2 [0;t
f
]: (3c)
We deﬁned the acceleration and jerk constraints in a sim-
ilar way as [8].
3 Optimal velocity proﬁle algorithm
The predeﬁned pathC in the two-dimensional space is
given as s
p
(u) = [x
p
(u);y
p
(u)]
T
with parameter u2
[u
0
;u
f
]. The parametrized velocity is:
v
p
(u) =
ds
p
(u)
du
=
  x
0
p
(u);y
0
p
(u)
  T
(4)
with the magnitude:
v
p
(u) =kv
p
(u)k=
q
(x
0
p
(u))
2
+(y
0
p
(u))
2
: (5)
The orientation equals the four-quadrant inverse tangent
of the quotient of Cartesian components of the transla-
tional velocity:
  p
(u) = atan2(y
0
p
(u);x
0
p
(u)); (6)
from which follows the expression for the magnitude of
parametrized angular velocity:
!
p
(u) =
d 
p
(u)
du
=
x
0
p
(u)  y
00
p
(u)  x
00
p
(u)  y
0
p
(u)
x
02
p
(u)+y
02
p
(u)
:
(7)
The curvature  p
(u) is:
  p
(u) =
d 
p
(u)
dl
=
d 
p
(u)
du
  du
dl
=
=
x
0
p
(u)  y
00
p
(u)  y
0
p
(u)  x
00
p
(u)
(x
0
p
(u)
2
+y
0
p
(u)
2
)
3=2
;
(8)
wherel is the arc length parameter of a curve.
3.1 Step 1: Complying with velocity and accelera-
tion constraints
In order to apply actual velocity and acceleration restric-
tions, the two operating physical quantities ought to be
expressed as functions of time.
The robot’s movement on a path can be described by
monotonously increasing parameteru of the curve or by
timet; the former measuring the position on a trajectory
and the latter the time at which a certain position on a
path is reached. We present the dependence ofu ont by
scheduleu(t) and as a result the parametrized functions
v
p
, !
p
,   p
deﬁned in Eq. (5, 7, 8) become composite
functionsv
p
(u(t)),!
p
(u(t)) and  p
(u(t)). Applying the
chain rule allows us to calculate the magnitudesv(t) and
!(t):
v(t) =
      ds
p
(u(t))
du
  du
dt
      =v
p
(u(t))  _ u(t); (9a)
!(t) =
d 
p
(u(t))
du
  du
dt
=!
p
(u(t))  _ u(t): (9b)
Expressing the velocity and angular velocity solely as
functions of time reveals that the time dependant veloci-
ties differ from the corresponding parametrized ones by a
factor of _ u(t). Obtaining the desired velocity proﬁle thus
requires calculating the schedule u = u(t) and its time
derivative _ u(t). The curvature, unlike the velocity or an-
gular velocity, does not depend on the parametrization of
the curve:
  (t) =  p
(u(t)): (10)
Acceleration vector is the derivative of velocity vector
from Eq. (1a):
a(t) =
dv(t)
dt
= _ v(t)  ^
T+v(t)  _
^
T: (11)
The time derivatives of unit tangential and unit normal
vector can be expressed from the Frenet–Serret formulas:
_
^
T(t) =  (t)  v(t)  ^
N(t); (12a)
_
^
N(t) =    (t)  v(t)  ^
T: (12b)
Using the equality from Eq. (12a) we ﬁnd that:
a(t) = _ v(t)  ^
T(t)+  (t)  v
2
(t)  ^
N(t): (13)
The expressions for the normal and tangential compo-
nents with respect to time follow from Eqs. (1a, 9a, 13):
a
T
(t) =v
0
p
(u(t))  _ u(t)+v
p
(u(t))   u(t); (14a)
a
R
(t) =  p
(u(t))  v
2
p
(u(t))  _ u
2
(t): (14b)
The proposed method of incorporating speed and accel-
eration constraints in the calculation of velocity proﬁle,
indirectly determined by schedulesu(t) and _ u(t), begins
with the identiﬁcation ofN points on the curve, the so-
called turning points, where the curvature reaches the lo-
cal maximum. The value of parameteru in thei  th turn-
ing point (i =f1;:::;Ng) is denoted asu
TPi
. The speed
in these points reaches local minimum, the tangential ac-
celerationa
T
(t) is therefore zero and radial acceleration
a
R
(t) is maximal. From Eq. (14b) it follows:
_ u
TPi
=
r
a
R
MAX
v
2
p
(u
TPi
)    p
(u
TPi
)
: (15)
It is also possible to implement the initial and ﬁnal speed
requirements by treating the initial and ﬁnal point of the
trajectory similarly as the turning points and using Eq.
(9a). For values of _ u we get:
_ u
TP0
=
vj
t=0
v
p
(u
TP0
)
; _ u
TP
N+1
=
vj
t=t
f
v
p
(u
TP
N+1
)
; (16)
where the initial and the ﬁnal point are denoted asTP
0
andTP
N+1
, respectively.
To get the complete velocity proﬁle v(t) of the mo-
bile robot, the values of u and _ u have to be determined
also in between the turning points. Our realization of the
proposed method for this calculation stems from knowing
the ﬁxed values ofu
TPi
and _ u
TPi
fori2f0;1;:::;N +
72
1g and the analytical formula for  u as a function ofu and
_ u that follows from Eqs. (14a, 14b, 3b, 5):
 u(t) =  aT
MAX
s
1
x
02
p
+y
02
p
  (x
02
p
+y
02
p
)  2
p
a
2
R
MAX
_ u
4
(t)
  x
0
p
x
00
p
+y
0
p
y
00
p
x
02
p
+y
02
p
_ u
2
(t);
(17)
where all the quantities with the indexp depend directly
onu. The basic idea of the algorithm that returns sched-
ulesu(t) and _ u(t) is to ﬁnd the solution to the initial value
problem by applying a numerical method of solving ordi-
nary differential equations; or more speciﬁcally: to inte-
grate backward and forward in time around each turning
point where the initial conditionsu
TPi
and _ u
TPi
are set
in order to determine the discrete values of t, u and _ u.
According to Euler’s method (the simplest explicit itera-
tive method) the valuesu
k+1
and _ u
k+1
in the(k+1)-th
step of the calculation are:
_ u
k+1
= _ u
k
   u
k
  T
s
; (18a)
u
k+1
=u
k
  _ u
k
  T
s
; (18b)
where T
s
is the sampling time and u
k
, _ u
k
,  u
k
the val-
ues calculated in thek-th step and perform as the current
initial values and/or slopes with negative or positive sign
(backward/forward integration). The sought-after sched-
ule _ u(u(t)) is deﬁned by the minimum of the separate
proﬁles around the turning points as shown in Figure (1).
Figure 1: Resulting _ u(u(t)) proﬁle (bold line) on a path
sp(u) = (cos(u);sin(2u));u2 [0;2  ] is determined by the
separate proﬁles _ u(u(t)) around theTPi (thin lines).
3.2 Step 2: Complying with jerk constraints
The resulting velocity proﬁle of the ﬁrst step of the cal-
culation is infeasible for an actual implementation on a
robot due to the sudden changes of acceleration. To ap-
ply jerk constraints to the existing velocity proﬁle the ex-
pression from Eq. (1c) can be written more speciﬁcally
by differentiating acceleration vector in Eq. (13) using
Eqs. (12a, 12b):
j(t) =
da(t)
dt
=
   v(t)  v
3
(t)    2
(t)
  ^
T(t)
+
1
v(t)
    d
dt
v
3
(t)    (t)
  ^
N(t):
(19)
From Eqs. (9a, 19) we get the following expressions for
the tangential component of the jerk:
j
T
(t) =v
00
p
(u(t))  _ u
3
(t)+3v
0
p
_ u(t)   u(t)
  v
3
p
(u(t))  _ u
3
(t)    2
p
(u(t))
+v
p
(u(t))  ...
u(t);
(20)
and its radial component:
j
R
(t) = 3v
0
p
(u(t))  v
p
(u(t))  _ u
3
(t)    p
(u(t))+
+3v
2
p
(u(t))    p
(u(t))  _ u(t)   u(t)+
+v
2
p
(u(t))  _ u
3
(t)    0
p
(u(t)):
(21)
The third time derivative of parameter u(t) with imple-
mented jerk constraints follows from Eq. (20):
...
u =
1
v
p
    j
T
(t)  v
00
p
_ u
3
  3v
0
p
_ u u+v
3
p
_ u
3
  2
p
  : (22)
Using Eq. (22) and Eq. (3c) to determine the value
of j
T
(t), the aim of the second step of calculation is to
smooth the intervals in the original velocity proﬁle that
contain points with abrupt changes of acceleration. Simi-
larly as in the Eq. (18a, 18b), forward Euler integration is
applied, yet with an additional calculation in the(k+1)-
th step:
 u
k+1
=  u
k
+
...
u
k
  T
s
: (23)
To determine the adequate initial value of Euler method,
let us ﬁrst introduceu
CP
‘
and its corresponding time deriva-
tive _ u
CP
‘
, ‘2f1;:::;Mg, as the value of u and _ u in
the critical points on the curve, where the acceleration is
discontinuous. Two guesses for the initial value ofu for
each critical point CP
‘
, u
L
‘
and u
H
‘
, are then selected
according to the following restriction:
u
CP
‘  1
<u
L
‘
<u
H
‘
<u
CP
‘
: (24)
Euler integration from either u
H
‘
and u
L
‘
onward pro-
duces a smooth extension to the original proﬁle _ u(u);
in the former case the extension forms an obtuse angle
at point of intersection with the original proﬁle (Figure
(2), left) and in the latter case the extension never recon-
nects back onto the original proﬁle (Figure (2), middle).
The correct initial value for the forward Euler method is
found by bisecting the interval [u
L
‘
;u
H
‘
]. The solution
is a range of values of u and _ u that does not introduce
additional discontinuities when inserted into the existing
_ u(u) proﬁle (Figure 2, right).
Figure 2: A display of iteration steps of bisection for determi-
nation of a smooth _ u(u(t)) proﬁle that eliminates acceleration
discontinuities.
73
4 Results
In order to demonstrate our proposed method, the prob-
lem is deﬁned as follows: Compute the optimal veloc-
ity proﬁle that will result in the shortest travelling time
for a pathx
p
(u) = cos(u),y
p
(u) = sin(2u) whereu2
[0;2  ] with the following restrictions:v
MAX
= 1:5m=s,
a
T
MAX
= 2m=s
2
, a
R
MAX
= 4m=s
2
, j
T
MAX
= 10m=s
3
andj
R
MAX
= 10m=s
3
.
Figure (3) shows _ u(u) dependence after the ﬁrst (_ u
I
(u))
and the second step (_ u
II
(u)) of the calculation of the op-
timal velocity proﬁle. The velocity proﬁlev
I
(t) in Figure
(4) respects speed and acceleration restrictions, but the
ﬁnal velocity proﬁle v
II
(t) complies also with the jerk
constraints. Figure (5) shows the temporal dependence of
normalized values of speed, acceleration and jerk restric-
tions and proves optimality because at any given moment
on the curve one of these dynamical quantities reaches its
maximum.
Figure 3: _ uI(u) and _ uII(u) with identiﬁedTPi andCPi. The
hatched areas represent speed restriction intervals, shaded areas
jerk restriction intervals and areas with the white background
acceleration restriction intervals.
Figure 4:vI(t) andvII(t) proﬁles.
Figure 5: Normalized dynamical restrictions over time.
5 Conclusion
The results of this study support the proposed idea of gen-
erating an optimal velocity proﬁle for a wheeled mobile
robot that moves on an arbitrary path as in every point
in time of the movement the robot has either maximum
allowed speed, acceleration or jerk.
The present study has only investigated and limited
jerk at the switch-overs from the intervals of restricted
speed movement to the intervals of restricted accelera-
tion movement and vice versa, where the inﬁnite delta
impulses of the jerk were expected. Generally speaking,
the non-compliance to the jerk limitations could also ap-
pear in other points or intervals of the path if onlyj
T
MAX
and j
R
MAX
were set low enough. We are now in the
process of investigating this problem with signiﬁcantly
greater complexity.
Acknowledgements
The authors acknowledge the ﬁnancial support from the
Slovenian Research Agency (research core funding No.
P2-0219).
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