On Model Predictive Path Following and Trajectory
Tracking for Industrial Robots
Mathias Hauan Arbo∗, Esten Ingar Grøtli† and Jan Tommy Gravdahl∗
∗Department of Engineering Cybernetics
NTNU, Norwegian University of Science and Technology
†Mathematics and Cybernetics, SINTEF DIGITAL, Trondheim, Norway
Abstract—In this article we show how the model predictive
path following controller allows robotic manipulators to stop at
obstructions in a way that model predictive trajectory tracking
controllers cannot. We present both controllers as applied to
robotic manipulators, simulations for a two-link manipulator
using an interior point solver, consider discretization of the
optimal control problem using collocation or Runge-Kutta, and
discuss the real-time viability of our implementation of the model
predictive path following controller.
I. INTRODUCTION
Modern industrial robots weld, grind, screw, measure, film,
paint, pick and place, and perform other tasks that require the
robot to follow some geometric path in space. For a typical
robot cell, we simplify our work as robotics engineers by
having enclosed, structured workspaces with no obstructions.
For dedicated large-scale production of a small set of products,
this is easily achieved. For small-scale manufacturing however,
there is a large variety of products and each product series
is produced in low-scale or on-demand. This calls for rapid
prototyping of both the paths the robot has to move, and
its environment. We must consider control strategies that
can handle both unknown and known obstructions without
sacrificing quality of the product.
We consider model predictive control (MPC), where we
may define constraints on states, and rudimentary obstructions.
This article considers known obstructions, and focuses on the
difference between path-following and trajectory tracking.
In trajectory tracking model predictive control (TT-MPC),
the robot is to follow a path with an explicit path-timing.
The trajectory may even incorporate constraints on the torques
or velocities of the robot through optimal control problem
(OCP) approaches such as [1], where a time-optimal path
timing law under constraints is generated. The path timing
law specifies the relation between the desired path and time,
while accounting for state constraints during the execution of
the path. The OCP is solved under the assumption that the
robot is moving along the path, and is an open-loop approach
to the constraint handling. In [2] this was extended to also
allow constraints on the acceleration and inertial forces at the
end-effector.
In [3], a model predictive path-following controller (MPFC)
that handles both path-timing and error from path in the same
OCP is described. In [4], the MPFC is shown to converge to
the path given terminal constraints without needing terminal
penalties. In [5] the MPFC is implemented on a KUKA
LWR IV robot, without end penalty or a terminal constraint.
This is done with the ACADO framework [6], which uses
a sequential programming method (SQP), iteratively solving
quadratic programs approximating the nonlinear program us-
ing the qpOASES active set solver.
In [7], a real-time MPFC scheme for contouring control
of an x-y table is described. Here a linear time varying
approximation of the dynamics is used to define a QP which
is solved using an active-set solver. In [8], this is implemented
on an x-y table, and the MPFC outperformed both a similarly
implemented TT-MPC, and the industry standard cascaded PI
controlled set-point controller operating at a higher sampling
frequency.
In [9], an MPFC is applied to a tower crane. The OCP
is solved using the gradient projection method, an indirect
method where Pontryagin’s Maximum Principle is solved
for the OCP without inequality constraints on the states.
Instead slack variables are introduced to implicitly handle the
inequality constraints.
In this article we
• Draw attention to differences in MPFC and TT-MPC
behavior, with and without obstructions,
• compare the Runge-Kutta and collocation integration
method for the two strategies,
• solve the nonlinear programs (NLP) for the control strate-
gies using the interior point solver IPOPT,
• and discuss the framework for real-time applications.
II. THEORY
A. System
The robot is an n degrees-of-freedom system with the state-
space representation
˙y(t) = x(t)
(1a)
˙x(t) = fx(y(t), x(t), u(t))
(1b)
where y ∈Rn are the generalized coordinates, x ∈Rn are
generalized velocities, and u ∈R are the inputs. The function
fx describes acceleration. We assume the robot has known
forward kinematics, allowing us to define a point of interest
p on the robot such that
p(t) = fp(y(t))
(2)
arXiv:1703.02279v1  [cs.RO]  7 Mar 2017
where fp(·) ∈Rnp is found from the forward kinematics. We
describe the C1 reference path as ϱ(·) ∈Rnp, defined in the
same frame as p.
The path-timing variable s moves from 0 to sf. The TT-
MPC assumes s = t for t < sf and s = sf otherwise. The
MPFC controls s through the path-timing dynamics, which we
model as a double integrator
 ˙s(t)
¨s(t)

=
0
1
0
0
 s(t)
˙s(t)

+
0
1

v(t) ..= fs(s, ˙s, v),
(3)
with piecewise constant input v(·) ∈R. To ensure that we
never move backwards along the path, and that we have a
maximum along path speed, we constrain ˙s ∈[0, ˙su]. For
more information on choice of the path-timing dynamics, we
refer the reader to [4].
For the MPFC we define the extended state ξ
=
[yT , xT , s, ˙s]T , input w = [uT , v]T , and dynamics
˙ξ(t) =


x(t)
fx(ξ(t), w(t)
fs(ξ(t), w(t)

= fξ(ξ(t), w(t)).
(4)
Similarly, for the TT-MPC we define the extended state χ =
[yT , xT ] with dynamics
˙χ(t) =

x(t)
fx(χ(t), u(t)

= fχ(χ(t), u(t)).
(5)
We define the deviation from the path as
epf(t) ..= fp(y(t)) −ϱ(s(t)).
(6)
for the MPFC, and
ett(t) ..= fp(y(t)) −ϱ(t)
(7)
for the TT-MPC. For s to converge to sf we also define the
path-timing error
es(t) ..= s(t) −sf.
(8)
B. Optimal Control Problem
With the previously defined dynamics and errors, we can
describe the OCP for the MPFC as
min
epf , ˙epf ,es,u,v
Z tk+T
tk
Jpf(τ, ¯ξ(τ), ¯w(τ))dτ
(9a)
s.t.:
˙¯ξ(τ) = fξ(¯ξ(τ), ¯w(τ))
(9b)
¯ξ(tk) = ξ(tk)
(9c)
¯ξ(τ) ∈[¯ξl, ¯ξu]
(9d)
hc(¯ξ(τ)) ≤0
(9e)
and for the TT-MPC as
min
ett, ˙ett,es,u
Z tk+T
tk
Jtt(τ, ¯χ(τ), ¯u(τ))dτ
(10a)
s.t.:
˙¯χ(τ) = fχ( ¯χ(τ), ¯u(τ))
(10b)
¯χ(tk) = χ(tk)
(10c)
¯χ(τ) ∈[ ¯χl, ¯χu]
(10d)
hc( ¯χ(τ)) ≤0
(10e)
where subscript u refers to the upper bounds on the states,
subscript l refers to the lower bound, and hc describes other
constraints such as obstacles in the path. The bar is to
differentiate internal states of the MPC and the actual system.
The cost integrands are defined as
Jpf(τ, ¯ξ(τ), ¯w(τ)) =1
2 ¯epf(τ)T Q¯epf + 1
2
˙¯epf(τ)T Qd ˙¯epf
+ 1
2 ¯u(τ)T R¯u(τ)
+ 1
2q¯es(τ)2 + 1
2r¯v(τ)2
(11)
for the MPFC and
Jtt(τ, ¯χ(τ), ¯w(τ)) =1
2 ¯ett(τ)T Q¯ett + 1
2
˙¯ett(τ)T Qd ˙¯ett
+ 1
2 ¯u(τ)T R¯u(τ)
(12)
for the TT-MPC. The matrices Q, Qd, and R are positive
definite. The scalars q and r are positive. We have included
the derivative of the path deviation to reduce oscillations.
Solving the OCPs can be done by an indirect approach
using Pontryagin’s Maximum Principle as done in [9], or
a direct approach as done in [7]. We will apply the direct
simultaneous approach, reformulating the OCP as an NLP by
discretizing the problem. The direct simultaneous approach is
most common in real-time applications, with existing software
support such as ACADO [6] and CasADI [10].
C. Nonlinear Program and Interior Point
In this section we only give the discretization of (9) as the
TT-MPC is similar and simpler. To discretize the OCP we use
two different integration methods: the 4th order Runge-Kutta
(RK4), and collocation based on Lagrange polynomials with
d Legendre points.
The control input is a piecewise continuous function, con-
stant on intervals of length δt, which is the length of our
timesteps. This gives us a horizon of length NT = T/δt.
With the simultaneous approach we use the integration method
between each time step, and constrain the result and next state
to be equal.
1) Runge-Kutta: Given a δt, RK4 [11] gives us the equation
¯ξk+1 = ¯ξk + δt
6 (k1 + 2k2 + 2k3 + k4) = F(¯ξk, ¯wk)
(13)
with
k1 = fξ(¯ξk, ¯wk),
(14a)
k2 = fξ

¯ξk + k1
δt
2 , ¯wk

,
(14b)
k3 = fξ

¯ξk + k2
δt
2 , ¯wk

,
(14c)
k4 = fξ
¯ξk + k3δt, ¯wk


.
(14d)
The resulting NLP is then
min
q
φ(q)
(15a)
s.t.:
fe(q) = 0
(15b)
he(q) ≤0,
(15c)
where q = [¯ξT
k , ¯wT
k , . . . , ¯ξT
k+NT −1, ¯wT
k+NT −1, ¯ξT
k+NT ]T , the
cost function is approximated with the rectangle method
φ(q) =
k+NT
X
j=k
δtJpf(tj, ¯ξj, ¯wj),
(16)
and
fe(q) =


¯ξk −ξ(tk)
¯ξk+1 −F(¯ξk, ¯wk)
...
¯ξk+NT −F(¯ξk+NT −1, ¯wk+NT −1)

.
(17)
The inequality constraints use (9d)-(9e) enforced on ¯ξi for
i = k, . . . , k + Nt.
2) Collocation: For the collocation method, with d collo-
cation points, we define j = 0, . . . , d Lagrange polynomials
Lj(˜τ) =
d
Y
r=0,r̸=j
˜τ −θr
θj −θr
(18)
where ˜τ ∈[0, 1], θ0 is 0, and the other θi are Legendre
collocation points. The approximation of the state trajectory
between tk and tk+1 is then
¯ξ(τ) =
d
X
j=0
Lj
τ −tk
δt

¯ξk,j, for τ ∈[tk, tk+1]
(19)
where ¯ξk,j are optimization variables included in q. Requiring
the derivatives to be equal on the collocation points, and the
simultaneous constraint to hold, we have
¯ξk+1,0 =
d
X
j=0
Lj(1)¯ξk,j
(20)
fξ(¯ξk,j, ¯wk) −1
δt
d
X
r=0
˙Lr(θj) = 0, for j = 1, . . . , d
(21)
where Lr(1) and
˙Lr(θj) are independent of tk and are
precomputed. This gives a similar structure to (15) with the
cost function evaluated with ¯ξi,0 for i = k, . . . , k + Nt.
We have chosen to evaluate (9d) at all states ¯ξk,j and the
nonlinear inequality constraints, (9e), at ¯ξk,0. This reduces
the computational burden, but allows the collocation points
¯ξk,j between tk and tk+1 to violate the nonlinear inequality
constraints. The optimization vector is of the form
q =[¯ξT
k,0, ¯ξT
k,1, . . . , ¯ξT
k,d, ¯wT
k ,
. . . , ¯ξT
k+NT −1,d, ¯wT
k+NT −1,
¯ξT
k+NT ,0, . . . , ¯ξT
k+NT ,d]T
(22)
and equality constraint function
fe(q) =


¯ξk,0 −ξ(tk)
¯ξk+1,0 −Pd
r=0 Lr(1)¯ξk,r
fξ(¯ξk,0, ¯wk) −Pd
r=0 ˙Lr(θ0)¯ξk,0
...
fξ(¯ξk,1, ¯wk) −Pd
r=0 ˙Lr(θ1)¯ξk,d
...


.
(23)
3) Interior point solver: Primal Interior point methods
consider NLPs of the form
min
˜q
φ(˜q) −µ
˜n
X
i=0
ln(˜qn)
(24a)
s.t.:
fe(˜q) = 0
(24b)
where ˜q includes slack variables to make he an equality
constraint, and µ defines the steepness of the barrier associated
with the slack variables. For large values of µ the ln term
will dominate and the solution will tend to the middle of
the feasible region. As µ decreases, φ will dominate and the
solution will move towards the optimal solution. Solving for
decreasing µ will converge to the solution of (15). Interior
point methods are difficult to warm-start, as a too low µ may
make certain slack variables prematurely small and cause slow
convergence. In the timing tests, we have not used warm start
as the initial states are random, but in the MPC implementation
we use the previous ¯ξ predictions as an initial guess.
We will use the interior point solver IPOPT [12], a primal-
dual interior point solver, solving (24) using the primal-dual
equations, see section 3.1 in [12]. Interior point methods have
consistent runtime with respect to problem size, allowing us
to potentially include more states with little effect on runtime.
Convergence of the MPFC is ensured using terminal sets
and penalties as constructed in [4] where an example is
given for the same system as ours with different parameters.
In this article we do not consider the terminal cost and
penalty. Reasoning being that most industrial machines have
the accuracy to move to a set-point sufficiently close to the
path, allowing us focus on the run time.
III. SIMULATION
We consider a two-link manipulator. This can easily be ex-
tended to 6 degrees-of-freedom, and results here are indicative
of the larger systems as interior point methods are consistent
with respect to the number of variables. The system was
implemented using Python and the CasADi framework [10].
CasADi allows us to define symbolic expressions for the
various equations in (9), and evaluate the derivatives using al-
gorithmic differentiation, e.g. for RK4, which may be difficult
to do by hand. The framework supports IPOPT [12].
A. System
y1
y2
Fig. 1: Two-link manipulator with 2 revolute joints.
The robot has two links of length L1 and L2, with link
masses M1 and M2 at Lc1 and Lc2, and masses m1 and m2
at the joints. The point of interest is the tip of the end effector.
The system is described by
˙y = ˙x
(25a)
˙x = M(y)−1 (u −C(y, x)x −G(y))
(25b)
with
M(y) =

a1 + a2 cos(y2)
1
2a2 cos(y2) + a3
1
2a2 cos(y2) + a3
a3

(26)
C(y, x) =
−1
2a2 sin(y2)x2
−1
2 sin(y2)(x1 + x2)
1
2a2 sin(y2)x1
0

(27)
G(y) =
g1 cos(y1) + g2 cos(y1 + y2)
g2 cos(y1 + y2)

,
(28)
where
a1 = I1 + I2 + m1L2
c1 + m2(L2
1 + L2
2),
(29)
a2 = 2m2Lc2L1,
a3 = m2L2
c2 + I2,
(30)
g1 = (Lc1(m1 + M1) + L1(m2 + M2))g,
(31)
g2 = Lc2(m2 + M2)g.
(32)
For brevity we give a1, a2, a3, g1, g2 ∈R in Table I, for
more information see [13]. The joint angles are defined as in
Fig.1. The maximum torques are 30 Nm, the timesteps are
δt = 0.01 s, and if not otherwise specified the horizon is
T = 0.20 s.
TABLE I: System parameters
Parameter
a1
a2
a3
g1
g2
Value
0.5578
0.2263
0.0785
17.0694
4.3164
In order to study obstacle avoidance, we include obstacles
oi as bounding circles with known radius roi and position poi.
Their inequality equations are
hoi =


fp(¯ξ(t)) −poi


 −r2
oi > 0.
(33)
TABLE II: MPC Parameters
Parameter
Q
Qd
R
q
r
MPFC
104I2×2
101I2×2
10−3I2×2
1
10−3
TT-MPC
104I2×2
101I2×2
10−3I2×2
In actual applications a vision system would bound detected
objects or people by a circle that the point of interest is not
to enter. When present we consider two obstacles with: ro1 =
0.02m, at po1 = [0.55, 0.75]T , and ro2 = 0.04m, at po2 =
[0.4, 0.4]T .
The reference path is a circle of radius 0.2 with center at
[0.55, 0.55]T .
B. Results
1) Moving to origin: In Fig.2 we see the Cartesian paths of
the Runge-Kutta TT-MPC and MPFC. The black dot is ϱ(0).
For this simulation we used maximum joint speeds of 0.5π
rad/s to exaggerate the differences. The set-point of the TT-
MPC moves gradually while the TT-MPC is approaching the
path. The MPFC on the other hand first approaches the origin
of the path, then moves along it. If q is large compared to
Q, we will move along the path faster than to the path. The
MPFC has come further with no difference in path deviation
as ˙¯s is greater than the rate of t, allowing it to move faster
along the path.
2) Obstacles: In Fig.3 two obstacles have been placed in
the path, and the speed constraints are removed. Both MPFC
and TT-MPC pass the first obstacle, but the MPFC stops at
the second. The second obstacle is too large, and the horizon
is not long enough to pass behind the obstacle. The MPFC
will decrease ˙¯s to zero, see t ≈2s in Fig.5, whereas the TT-
MPC will have a gradually increasing cost as the trajectory
set-point moves forward through the obstacle, forcing it around
the object.
When obstructed, there was a difference between the two
integration methods for the TT-MPC. We saw that the collo-
cation method left the TT-MPC path closer to the obstacles,
see Fig.4. The MPFC decreased ˙s upon approaching the first
object, at t ≈0.9s in Fig.5, and was not affected by integration
differences.
3) Timing: The simulations were performed on a Macbook
Pro with a 2.5 Ghz i7 CPU. Using the compilation feature of
CasADi we can create implementations that approach speeds
needed for real-time systems. To compare the timings of the
two integration methods we have performed a Monte-Carlo
simulation of the MPFC with uniformly distributed initial
positions in the upper right quadrant of the workspace. Box
plot of the solver using RK4 for varying horizon lengths is
given in Fig.6. In Fig.7 we give the same for the collocation
method.
CasADi gives timing statistics of the solver. Upon in-
spection it appears that the collocation method has faster
evaluation of the constraint functions, Hessian of the problem
Lagrangian, and generally fewer iterations, but the increased
optimization vector length makes the solver slower. In Table
III we give typical timings of the solver. The cost function
and cost gradient are not included as they were the same and
approximately 1 ms.
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
xp [m]
0.3
0.4
0.5
0.6
0.7
0.8
yp [m]
ref
MPFC
TT-MPC
Fig. 2: TT-MPC and MPFC moving from the same start point
towards the path. The blue dot is the start of the reference
path.
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
xp [m]
0.3
0.4
0.5
0.6
0.7
0.8
yp [m]
ref
obstacle
MPFC
TT-MPC
Fig. 3: TT-MPC and MPFC initialized close to the path origin.
The path is obstructed by a small and a large object. Both
controllers pass the first object, but the MPFC does not pass
the second.
TABLE III: Typical timing statistics T = 0.6
Constr.1
∇qConstr.
H(L)2
Iter.
Solver tot.
MPFC-RK4
0.03 ms
0.16 ms
0.52 ms
27
0.045 s
MPFC-Col
0.03 ms
0.08 ms
0.15 ms
24
0.102 s
TT-MPC-RK4
0.03 ms
0.23 ms
0.66 ms
14
0.039 s
TT-MPC-Col
0.03 ms
0.07 ms
0.11 ms
14
0.046 s
IV. DISCUSSION AND FUTURE WORK
The collocation method has a sparse structure in the equality
constraints, and relies on evaluation of fξ, whereas the RK4
1The constraint function include both the inequality constraints and the
ODE dynamics.
2Hessian of the problem Lagrangian.
0.54
0.55
0.56
0.57
0.58
0.59
0.60
xp [m]
0.73
0.74
0.75
0.76
0.77
yp [m]
ref
obstacle
MPFC
TT-MPC
(a) Collocation method
0.54
0.55
0.56
0.57
0.58
0.59
0.60
xp [m]
0.73
0.74
0.75
0.76
0.77
yp [m]
ref
obstacle
MPFC
TT-MPC
(b) RK4
Fig. 4: Difference between the two integration methods when
the path is obstructed. Only evident in TT-MPC as the MPFC
slows sufficiently down before the obstacle to not encounter
the integration problems.
0
1
2
3
4
5
6
t [s]
0
1
2
3
4
5
6
path-timing
s
˙s
sf
Fig. 5: Timing parameter s, blue line, and ˙s, green line, when
MPFC follows the obstructed path. First object encountered at
t ≈0.9s, second object encountered at t ≈2s.
requires evaluation of its algorithimic differentiation. This
gives the collocation method faster function evaluations, but
it did not appear to be sufficient to make the collocation
method faster than RK4. The solver itself took more time
with the increased number of states. For systems with complex
dynamics the collocation method may be necessary. TT-MPC
had fewer states, simpler dynamics and was faster on the
4
8
12
16
20
24
Horizon lengths [timesteps]
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050
runtime [s]
Fig. 6: Boxplot of 2000 RK4 MPFC solver run times.
4
8
12
16
20
24
Horizon lengths [timesteps]
0.00
0.02
0.04
0.06
0.08
0.10
runtime [s]
Fig. 7: Boxplot of 2000 collocation MPFC solver run times.
whole, with the same observations as for the MPFC regarding
integration method.
The MPFC has the ability to stop along its path. For time-
critical systems this is not desired, but for robots in open
environments it can prove useful. It also suggests that when
obstructed by an unknown object, it may push against the
object with a constant force. In future experiments this will
be investigated further. The TT-MPC will observe a growing
difference in the current position and the desired position. With
a known obstruction it will project the path onto the constraint
attempting to minimize the path error. Suddenly removing
the obstruction should result in the TT-MPC moving towards
its current set point as fast as possible. With an unknown
obstruction the TT-MPC may exert a gradually increasing
force on the obstruction.
For the MPC to be real-time feasible, we require the solver
to run faster than the control interval used. In this article
we have considered a control interval of length δt = 0.01
s. For low horizon lengths we are approaching such timing
with the CasADi running in Python. Future work will extend
this framework for a 6 degrees-of-freedom robot with a 3D
path. The low horizon length needed to be able to achieve fast
run time of the solver suggests that terminal constraints may
be required in the final system.
The obstructions considered in this article were static and
known apriori. Future work may include varying number of
obstacles that enter the robot workspace.
V. CONCLUSION
The model predictive path-following controller gives rise to
a set of new design opportunities. Of most value for obstructed
environments is the fact that it may freely stop and resume
along its path. The question is whether a constraint ends the
path, as the path-following controller did, or whether the robot
should move along the path projected onto the constraint, as
the trajectory tracking controller did.
We also saw that the interior point method of IPOPT
interfaced through CasADi in Python, approached timings we
would desire in a real-time systems.
VI. ACKNOWLEDGEMENTS
The work reported in this paper was based on activities
within centre for research based innovation SFI Manufacturing
in Norway, and is partially funded by the Research Council
of Norway under contract number 237900.
REFERENCES
[1] D. Verscheure, B. Demeulenaere, J. Swevers, J. De Schutter, and
M. Diehl, “Time-optimal path tracking for robots: A convex optimization
approach,” IEEE Transactions on Automatic Control, vol. 54, no. 10, pp.
2318–2327, 2009.
[2] F. Debrouwere, W. Van Loock, G. Pipeleers, M. Diehl, J. Swevers, and
J. De Schutter, “Convex time-optimal robot path following with Carte-
sian acceleration and inertial force and torque constraints,” Proceedings
of the Institution of Mechanical Engineers, Part I: Journal of Systems
and Control Engineering, vol. 227, no. 10, pp. 724–732, nov 2013.
[3] T. Faulwasser, B. Kern, and R. Findeisen, “Model predictive path-
following for constrained nonlinear systems,” Proceedings of the 48h
IEEE Conference on Decision and Control (CDC) held jointly with 2009
28th Chinese Control Conference, no. 3, pp. 8642–8647, 2009.
[4] T. Faulwasser and R. Findeisen, “Nonlinear Model Predictive Control for
Constrained Output Path Following,” IEEE Transactions on Automatic
Control, vol. 9286, no. c, pp. 1–1, 2016.
[5] T. Faulwasser, T. Weber, P. Zometa, and R. Findeisen, “Implementation
of Nonlinear Model Predictive Path-Following Control for an Industrial
Robot,” IEEE Transactions on Control Systems Technology, pp. 1–7,
2016.
[6] B. Houska, H. J. Ferreau, and M. Diehl, “ACADO Toolkit – An Open
Source Framework for Automatic Control and Dynamic Optimization,”
Optimal Control Applications and Methods, vol. 32, no. 3, pp. 298–312,
2011.
[7] D. Lam, C. Manzie, and M. Good, “Model predictive contouring
control,” in 49th IEEE Conference on Decision and Control (CDC),
vol. 86, no. 8.
IEEE, dec 2010, pp. 6137–6142.
[8] ——, “Application of model predictive contouring control to an X-Y
table,” IFAC Proceedings Volumes (IFAC-PapersOnline), vol. 18, no.
PART 1, pp. 10 325–10 330, 2011.
[9] M. Böck and A. Kugi, “Real-time nonlinear model predictive path-
following control of a laboratory tower crane,” IEEE Transactions on
Control Systems Technology, vol. 22, no. 4, pp. 1461–1473, 2014.
[10] J. Andersson, “A General-Purpose Software Framework for Dynamic
Optimization,” PhD thesis, Arenberg Doctoral School, KU Leuven,
Belgium, 2013.
[11] O. Egeland and J. T. Gravdahl, Modeling and Simulation for Automatic
Control.
Trondheim: Marine Cybernetics, 2003.
[12] A. Wächter and L. T. Biegler, On the Implementation of Primal-Dual
Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear
Programming, 2006, vol. 106, no. 1.
[13] A. Astolfi, D. Karagiannis, and R. Ortega, Nonlinear and Adaptive
Control with Applications (Communications and Control Engineering),
2007.