Stabilizing Receding Horizon Control for Three Dimensional
Visual Feedback System with Fixed Camera Configuration
Toshiyuki Murao and Masayuki Fujita
Abstract— This paper investigates vision based robot control based on a receding horizon control strategy, as a firststep for a predictive visual feedback control. Firstly, the briefsummary of the 3D dynamic visual feedback system with fixedcamera configuration is given. Next, a stabilizing receding
horizon control for the 3D dynamic visual feedback system,a highly nonlinear and relatively fast system, is proposed. The
stability of the receding horizon control scheme is guaranteedby using the terminal cost derived from an energy functionof the visual feedback system.
Keywords— Visual Feedback Control, Lyapunov Stability,
Passivity, Receding Horizon Control

Robot
Manipulator

T. Murao and M. Fujita are with Department of Mechanical andControl Engineering, Tokyo Institute of Technology, Tokyo 152-8552,
fujita@ctrl.titech.ac.jp

gch

5c

gwc
gco

5h
gwh

gho
5o

gwo

5w
World Frame

Target Object

I. I NTRODUCTION
Robotics and intelligent machines need sensory information to behave autonomously in dynamical environments. Visual information is particularly suited to recognize unknown surroundings. In this sense, vision is
one of the highest sensing modalities that currently exist.
Vision based control of robotic systems involves the fusionof robot kinematics, dynamics, and computer vision tocontrol the motion of the robot in an efficient manner. The
combination of mechanical control with visual information,so-called visual feedback control or visual servoing, isimportant when we consider a mechanical system working
in dynamical environments [1], [2].
In previous works, for the problem of three dimensional(3D) visual servo control, Kelly et al. [3] considered
an image based controller under the assumption that theobjects’ depths are known. Cowan et al. [4] addressed
the field-of-view problem for 3D dynamic visual feedback
system using navigation functions. Although good solutions to the set-point problem are reported in those papers,
few results have been obtained for the tracking problem
of moving target objects in the full 3D dynamic visualfeedback system that include not only the position and theorientation but also the manipulator dynamics. The authors
discussed passivity based control for a moving target object
in 3D workspace with eye-in-hand configuration [5], [6]and fixed camera configuration [7]. However, the control
law proposed in [5]–[7] is not based on optimization,
the desired control performance cannot be guaranteed
explicitly.
Receding horizon control, also recognized as model
predictive control is a well-known control strategy inwhich the current control action is computed by solving, a

Image

CameraFig. 1.

Visual feedback system with fixed camera configuration.

finite horizon optimal control problem on-line [8]. A large
number of industrial applications using model predictive
control can be found in chemical industries where theprocesses have relatively slow dynamics. On the contrary,for nonlinear and relatively fast systems such as in robotics,
few implementations of the receding horizon control havebeen reported. For the receding horizon control, many
researchers have tackled the problem of stability guarantees. An approach proposed by Parisini et al. [9] isbased on using a quadratic endpoint penalty of the formaxT (t + T )P x(t + T ) for some a > 0, some positive
definite matrix P and a terminal state x(t + T ). Jadbabaie
et al. [10] showed that closed-loop stability is ensured
through the use of a terminal cost consisting of a control
Lyapunov function. Moreover, these results were applied to
the Caltech Ducted Fan to perform aggressive maneuvers[11], [12]. Visual feedback, however, is not consideredhere. Predictive control could be of significant benefit
when used in conjunction with visual servoing. With
the incorporation of visual information, the system could
anticipate the target’s future position and be waiting there
to intercept it [13]. In [14], the authors proposed stabilizing
receding horizon control for the planar visual feedback
system. Moreover, for the 3D dynamic visual feedback system with eye-in-hand configuration, a stabilizing receding
horizon control is derived in [15]. But the 3D dynamic
visual feedback system has only three coordinate frames,
while visual feedback systems typically use four coordinate
frames as in Fig. 1. Because the camera is attached tothe end-effector of robots, the camera frame represents thehand one in eye-in-hand configuration. This may occur thatthe visual feedback systems cannot be represented in eyein-hand configuration, such as the autonomous injection of

biological cells [16] and so on.
In this paper, as a first step for a predictive visualfeedback control, stabilizing receding horizon control isapplied to the 3D visual feedback system with fixed camera
configuration in [7], a highly nonlinear and relatively fastsystem. Compared with the previous work [15], because
the passivity-based visual feedback system with eye-inhand configuration [5] can be regarded as the special
case of that with fixed camera configuration [7], then thepossible application area should be undoubtedly increasing.
This represents a first step towards high performance visual
servoing targeting more aggressive maneuvers. The mainidea is the use of the terminal cost derived from an energyfunction of the visual feedback system.
Firstly, the brief summary of our prior work [7], which isthe 3D dynamic visual feedback system with fixed camera
configuration, is given. Next, a stabilizing receding horizon
control for the 3D visual feedback system using a control
Lyapunov function is proposed.
Throughout this paper, we use the notation eξ̂θab ∈3×3
R
to represent the change of the principle axes ofa frame Σb relative to a frame Σa . ξab ∈ R3 specifies
the direction of rotation and θab ∈ R is the angle ofˆ ab to denote ξˆab θab . The
rotation. For simplicity we use ξθ
notation ‘∧’ (wedge) is the skew-symmetric operator suchthat ξ̂θ = ξ × θ for the vector cross-product × and any
vector θ ∈ R3 . The notation ‘∨’ (vee) denotes the inverse
operator to ‘∧’, i.e., so(3) → R3 . Recall that a skewsymmetric matrix corresponds to an axis of rotation (via
the mapping a → â). We use the 4 × 4 matrixeξ̂θab pab
gab =
(1)
0
1
)∈
as the homogeneous representation of gab = (pab , e
SE(3) describing the configuration of a frame Σb relative
to a frame Σa . The adjoint transformation associated withgab is denoted by Ad(gab ) [17].
ξ̂θabII. PASSIVITY- BASED V ISUAL F EEDBACK S YSTEM
WITH F IXED C AMERA C ONFIGURATIONIn this section, the brief summary of our prior work in[7] is given. An energy function and a stabilizing control
law, which play an important role for a predictive visualfeedback control, are derived.
A. Basic Representation for Visual Feedback System
Visual feedback systems typically use four coordinate
frames which consist of a world frame Σw , a target object
frame Σo , a camera frame Σc and a hand (end-effector)
frame Σh as in Fig. 1. Then, gwh , gwc and gwo denote
the rigid body motions from Σw to Σh , from Σw to Σcand from Σw to Σo , respectively. Similarly, the relative
rigid body motions from Σc to Σh , from Σc to Σo andfrom Σh to Σo can be represented by gch , gco and gho ,respectively, as shown in Fig. 1. The objective of visualfeedback control is to bring the actual relative rigid body
motion gho to a given reference gd . The reference gd isassumed to be constant throughout this paper, because theend-effector of manipulator (we call hand) can track themoving target object in this case. Our goal is to determine
the hand’s motion using the visual information for thispurpose.
The relative rigid body motion from Σc to Σo can be
led by using the composition rule for rigid body transformations ([17], Chap. 2, pp. 37, eq. (2.24)) as follows:−1
gco = gwc
gwo .

(2)

The relative rigid body motion involves the velocity ofeach rigid body. To this aim, let us consider the velocity
of a rigid body as described in [17]. We define the body
velocity of the camera relative to the world frame Σwb
T
T T
as Vwc= [vwcωwc] , where vwc and ωwc represent thevelocity of the origin and the angular velocity from Σw toΣc , respectively ([17] Chap. 2, eq. (2.55)).
Differentiating (2) with respect to time, the body velocity of the relative rigid body motion gco can be written as
follows [7].
b
b
b
−1 V
Vco
= −Ad(gco) wc + Vwo(3)bis the body velocity of the target object relative
where Vwo
to Σw . We consider that it is the basic representation for
the three coordinate frames of the visual feedback system.
Roughly speaking, the relative rigid body motion gco willb
depend on the difference between the camera velocity Vwcbb
and the target object velocity Vwo , because Vco is definedas the body velocity of the relative rigid body motion gco .
b
In the case of fixed camera configuration, i.e. Vwc= 0,b
the fundamental representation Vco can be rewritten as
b
Vco

b
= Vwo
.

(4)

b
equals the target
Hence, the fundamental representation Vcobobject velocity Vwo , in the case of fixed camera configuration.

B. Estimation Error and Control Error Systems
The visual feedback control task requires information of
the relative rigid body motion gco . Since the measurable
information is only the image information f (gco ) in thevisual feedback system, we consider a nonlinear observer
in order to estimate the relative rigid body motion fromthe image information f (gco ).
Firstly, using the basic representation (4), we choosebestimates ḡco and V̄coof the relative rigid body motion
and velocity, respectively as
b
V̄co

=

ue

(5)

where ue is the new input in order to converge theestimated value to the actual relative rigid body motion.
In order to establish the estimation error system, we
define the estimation error between the estimated valueḡco and the actual relative rigid body motion gco as−1
gco .
gee = ḡco(6)

Using the notation eR (eξ̂θ ) in [18], the vector of theestimation error is given by ee := [pTee eTR (eξ̂θee )]T . Notethat ee = 0 iff pee = 0 and eξ̂θee = I3 . Therefore, if
the vector of the estimation error is equal to zero, thenthe estimated relative rigid body motion ḡco equals theactual relative rigid body motion gco . The estimation error
vector ee can be obtained from image information f (gco )and the estimated value of the relative rigid body motionḡco . In the same way as the basic representation for thevisual feedback system, the estimation error system canbe represented bybb−1 ue + V
= −Ad(gee
Vee
wo .
)

(7)

Moreover, let us consider the dual of the estimation error
system, which we call the control error system, in order toestablish the visual feedback system. We define the control
error between gd and ḡho , which is called the control error,
as follows
gec = gd−1 ḡho ,(8)where ḡho is the estimated relative rigid body motion from−1
Σh to Σo and obtained from ḡho = gchḡco . Here, we
assume that gch is calculated by using the known motion,i.e. gwc and gwh , exactly. The vector of the control error isdefined as ec := [pTec eTR (eξ̂θec )]T . The control error system
is described bybb= −Ad(ḡ−1 ) Vwh
+ ue
Vec
ho

(9)

b
where Vwh
is the body velocity of the hand relative to Σw .
Combining (7) and (9), we construct the visual feedback
system with fixed camera configuration as follows:  b 
I−Ad(ḡ−1 )
0 b
Vec
ho
Vwo (10)
uce +
b =
I
Vee
−1
0
−Ad(gee )b T) uTe ]T denotes the control input.
where uce := [(VwhHere, if the following condition (11) is satisfied, thepassivity-based visual feedback system with eye-in-handconfiguration [5] can be regarded as the special case ofthat with fixed camera configuration [7],
b
b
= Vwc
.
gch = I, Vwh(11)

C. Dynamic Passivity based Visual Feedback System with
Fixed Camera Configuration
The manipulator dynamics can be written as
M (q)q̈ + C(q, q̇)q̇ + g(q) = τ + τd(12)where M ∈ Rn×n is the inertia matrix, C ∈ Rn×n is the
Coriolis matrix, g ∈ Rn is the gravity terms, and q, q̇ and q̈
are the joint angle, velocity and acceleration, respectively.
τ is the vector of the input torque, and τd represents a
disturbance input [19]. Now, we will construct a dynamic
visual feedback system by connecting the visual feedback
system (10) and the manipulator dynamics (12). The bodybis given byvelocity of the hand VwhbVwh= Jb (q)q̇(13)

where Jb (q) is the manipulator Jacobian [17].
Next, we propose the control law for the manipulator asτ =M (q)q̈d +C(q, q̇)q̇d +g(q)+JbT (q)AdT(g−1 ) ec +uξ . (14)dwhere q̇d and q̈d represent the desired joint velocity andacceleration, respectively. The new input uξ is to be
determined in order to achieve the control objective.
Let us define the error vector with respect to the joint
velocity of the manipulator as ξ := q̇ − q̇d . Moreover,
we design the reference of the joint velocity based on therelation between the hand velocity and the joint velocity(13) as q̇d := Jb† (q)ud where ud is the desired body
velocity of the hand which will be obtained from the visualbin (9) should be replaced by
feedback system. Thus, Vwh
ud .
Using (10)–(14), the visual feedback system with themanipulator dynamics with fixed camera configuration (wecall the dynamic visual feedback system with fixed camera
configuration) can be derived as follows:⎤⎤ ⎡⎡−M −1 Cξ + M −1 JbT AdT(g−1 ) ec
ξ̇
d
⎥
b ⎦=⎢
⎣ Vec
−Ad(ḡ−1 ) Jb ξ
⎦
⎣
ho
b
Vee
0
⎡ −1
⎤ ⎡ −1 ⎤M00M0⎦u+⎣ 0 0 ⎦w (15)−1+⎣ 0 −Ad(ḡho
) I
0 I
−1
0
0 −Ad(gee)
where u := [uTξ uTd uTe ]T . We define the state and the
disturbance of dynamic visual feedback system as x :=b T T
[ξ T eTc eTe ]T and w := [τdT (Vwo) ] , respectively.
D. Energy Function and Stabilizing Control Law
In previous work [5], the passivity of the visual feedback
system (15) is derived by using the following energyfunction V (x)V (x) =1 T
ξ M (q)ξ + E(gec ) + E(gee )2(16)where E(g) := 12 p2 +φ(eξ̂θ ), and φ(eξ̂θ ) := 12 tr(I−eξ̂θ )is the error function of the rotation matrix (see e.g. [18]).
Here, we consider the following control input
⎤
⎡
0
Kξ 0
u = −Kν := uk , K := ⎣ 0 Kc 0 ⎦ ,(17)
0
0 Ke
⎡
⎤
I
0
0
T
⎢
⎥
ν := N x := ⎣ 0 −Ad(gd−1 ) 0 ⎦ x.
(18)
T
0 Ad(eξ̂θec ) −I
where Kξ:=
diag{kξ1 , · · · , kξn }∈Rn×n ,
6×6
and Ke :=Kc := diag{kc1 , · · · , kc6 } ∈ R
diag{ke1 , · · · , ke6 } ∈ R6×6 are positive gain matrices.
By differentiating V (x) along the trajectory of the system
and using the control input uk , the next equation is
obtained if w = 0 is satisfied.
V̇ = ν T u = −xT N T KN x.

(19)Visual Feedback System

N

u

Vecb
Veeb

+

u

x

Reference

OutputPredictive Output
Predictive Input

Passivity-based Control Law(Stabilizing Control Law)uk
Fig. 2.

Control Input

KNx

t +T

t
Fig. 3.

Block diagram of the passivity based visual feedback control.

Therefore, if w = 0, the equilibrium point x = 0 for theclosed-loop system (15) and (17) is asymptotic stable, i.e.,
uk is a stabilizing control law for the system. The block
diagram of the passivity based visual feedback control isshown in Fig. 2.
However, the stabilizing control law uk (17) is not basedon optimization, the desired control performance cannot
be guaranteed explicitly. In the next section, a stabilizing
receding horizon control based on optimal control theory
is proposed.
III. S TABILIZING R ECEDING H ORIZON C ONTROL FOR
T HREE D IMENSIONAL V ISUAL F EEDBACK S YSTEM
WITH F IXED C AMERA C ONFIGURATIONThe objective of this section is to propose a predictive
visual feedback control based on optimal control theory.
A camera can provide more information than the current
derivation from a nominal position at the sample instant.
This property can be exploited to predict the target’s future
position and improve the control performance. As a firststep for a predictive visual feedback control, we propose astabilizing receding horizon control based on optimizationin this paper.

Receding horizon scheme.

The following lemma concerning a control Lyapunov
function is important to prove a stabilizing receding horizon control. The definition for a control Lyapunov function
M (x) is given by
inf Ṁ (x) + l(x, u) ≤ 0,

(23)

u

where l(x, u) is a positive definite function [10].
Lemma 1: Suppose that the following matrix P is positive semi definite and w = 0.
P := ρN T KN − Q − N T K T RKN,

ρ>0Then, the energy function ρV (x) of the visual feedback
system (15) can be regarded as a control Lyapunov function.
Proof: In [5], we have already shown that the time
derivative of V along the trajectory of the system (15)
is formulated as (19). Using the positive definite functionl(x(t), u(t)) (21) and the stabilizing control law uk (17)for the system, Equation (23) can be transformed into

A. Control Lyapunov Function

inf [Ṁ (x) + l(x, u)]

In this section, the finite horizon optimal control problem (FHOCP) for the visual feedback system (15) is considered. The FHOCP at time t consists of the minimization
with respect to the input u(τ, x(τ )), τ ∈ [t, t + T ], of thecost function= inf ρV̇ + xT Qx + uT Ru
u = inf ρxT N T u + xT Qx + uT Ru

ut+T

J(x0 , u, T )=l(x(τ ), u(τ ))dτ + F (x(t + T )) (20)
t
T

(21)
l(x(t), u(t)) = x (t)Q(t)x(t) + uT (t)R(t)u(t).
Q(t)≥0, R(t) > 0, F (x(t + T ))≥0with the state x(t) = x0 . For a given initial condition x0 ,we denote this solution of the FHOCP as u∗ (τ, x(τ )), τ ∈[t, t+T ]. In receding horizon control, at each sampling timeδ, the resulting feedback control at state x0 is obtained bysolving the FHOCP and settinguRH := u∗ (0, x0 ).
Fig. 3 shows the receding horizon scheme.

(24)

(22)

u

= −xT (ρN T KN − Q − N T K T RKN )x= −xT P x.

(25)

This completes the proof.
Lemma 1 shows the energy function ρV (x) of thevisual feedback system (15) can be regarded as a control
Lyapunov function in the case of P ≥ 0. This assumption
P ≥ 0 is satisfied if ρ is chosen large enough.
B. Stabilizing Receding Horizon Control for the 3D VisualFeedback System with fixed camera configuration
Suppose that the terminal cost is the control Lyapunov
function ρV (x), the following theorem concerning thestability of the receding horizon control holds.

Theorem 1: Consider the following cost function for thevisual feedback system (15).
t+T

J(x0 , u, T )=l(x(t), u(t)) = x (t)Q(t)x(t) + uT (t)R(t)u(t)Q(t)≥0, R(t) > 0
F (x) = ρV (x). ρ > 0

where uk is the stabilizing control law (17) for the system.
The associated cost isJ(x∗ (t + δ), ũ, T )= J(x(t), u∗ , T ) + ρ[V (x(t + T + δ)) − V (x∗ (t + T ))]l(x∗ (τ ), u∗ )dτ +tt+T +δ

l(x∗ (τ + T ), uk )dτ,

t+T

(30)

where x∗ is the optimal state trajectory. This cost, which
is an upper bound for J(x∗ (t + δ), u∗ , T ), satisfies
J(x∗ (t + δ), u∗ , T ) − J(x∗ (t), u∗ , T )≤ ρ[V (x(t + T + δ)) − V (x∗ (t + T ))]
t+δ

−
t

l(x∗ (τ ), u∗ )dτ +t+T +δ

t+T

+

u

x

Receding Horizon Control Law(28)

Proof: Our goal is to prove that J(x∗ (t), uRH , T ),which is the cost-to-go applying the receding optimal
control uRH , will qualify as a Lyapunov function for theclosed loop system. Construct the following suboptimal
control strategy for the time interval [t + δ, t + T + δ] ∗u (τ ) τ ∈ [t + δ, t + T ]
(29)
ũ =
uk (τ ) τ ∈ [t + T, t + T + δ]

t+δ

N

Vecb
Veeb

q Angle(27)

Suppose that P (24) is positive semi definite and w = 0,then the receding horizon control for the visual feedback
system is asymptotically stabilizing.

−ul(x(τ ), u(τ ))dτ + F (x(t + T ))(26)
t
T

Joint

Visual Feedback System

uRH
Fig. 4.

Block diagram of the predictive visual feedback control.

horizon control design is based on optimal control theory,the control performance should be improved compared tothe simple passivity based control [7], under the condition of adequate gain assignment in the cost function.
Moreover, compared with previous work [15], the main
advantage of this approach is that the 3D dynamic visualfeedback system with fixed camera configuration can treat
not only three coordinate frames but also four coordinate
frames which is used for typically visual feedback systems.
This allows us to extend the technological application area.
In this paper, as a first step, we propose unconstrained
stabilizing receding horizon control schemes. In the near
future, we will consider constraints which represent one of
the advantages of receding horizon control, and develop
it using level set, see [10]. The block diagram of thepredictive visual feedback control is shown in Fig. 4.
Moreover, focused on the inverse optimality approach,the following corollary is derived.
Corollary 1: Consider the following weights of the costfunction (26)–(28).

l(x∗ (τ + T ), uk )dτ.
(31)

Using the positive definite function l(x(t), u(t)) (21) andthe stabilizing control law uk (17) for the system, anddividing both sides by δ and taking the limit as δ → 0,
Equation (31) can be transformed intoJ(x∗ (t + δ), u∗ , T ) − J(x∗ (t), u∗ , T )δ→0δ≤ −x∗T (t + T )(ρN T KN − Q − N T K T RKN )x∗ (t + T )−x∗T (t)Qx∗ (t) − u∗T Ru∗= −x∗T (t + T )P x∗ (t + T ) − x∗T (t)Qx∗ (t) − u∗T Ru∗ .
(32)
lim

Considering that the control input during first δ is
uRH = u∗ , by the assumption P ≥ 0, the derivativeof J(x∗ (t), uRH , T ) is negative definite. Therefore, wehave shown that J(x∗ (t), uRH , T ) qualifies as a Lyapunov
function and asymptotic stability is guaranteed.
Theorem 1 guarantees the stability of the receding
horizon control using a control Lyapunov function for the3D visual feedback system (15) which is a highly nonlinear
and relatively fast system. Since the stabilizing receding

Q(t) =
R(t) =
ρ=

qN T (t)KN (t) q≥0
rK −1 r > 0
√
2 qr

(33)
(34)
(35)

If w = 0, then the receding horizon control for the visualfeedback system is asymptotically stabilizing, the receding
horizon control law is

q
KN x
(36)
uRH = −rand the cost-to-go is given by
J(x0 , uRH , T ) = ρV (x).

(37)

Proof: As a preliminary, the following calculation
concerning the cost function which is set to (33)–(35) isneededḞ (x) + l(x, u)= ρV̇ + xT Qx + uT Ru√= 2 qrxT N T u + qxT N T KN x + ruT K −1 u
√
√
T

√
√
= ru + qKN x K −1 ru + qKN x

√
√
2
(38)
=  ru + qKN xK −1 .

JointVisual Feedback System

N

u

Vecb
Veeb

+

q Angleusystem. In this paper, the stabilizing receding controller
was implemented for a low level inner loop, in the near
future, we would like to tackle the implementation on ahigh level outer loop.

x

R EFERENCES
RHC Law (Inverse Optimal Approach)

uRH

q
r KNx

Fig. 5. Block diagram of the predictive visual feedback control focusedon the inverse optimality approach.

Using the above equation (38), the cost function (26) canbe transformed intot+T xT Qx + uT Ru dτ + ρV (x(t + T ))J(x0 , u, T )=tt+T √ ru + √qKN x2 −1 − ρV̇ dτ=Kt+ρV (x(t + T ))2√
 ru + qKN x −1 dτ + ρV (x(t)).
K

t+T √=t(39)

Thus, the receding horizon control lawquRH = −
KN x
r

(40)

minimizes the cost function (26), and the cost-to-go isgiven byJ(x0 , uRH , T ) = ρV (x).

(41)

By regarding the cost-to-go J(x0 , uRH , T ) as a Lyapunov
function and using the receding horizon control input (36),the time derivative of ρV along trajectory of the system(15) can be transformed into
˙ 0 , uRH , T ) = ρV̇ (x)
J(x

q T T
x N KN x.
= −ρ
r

(42)

Therefore, the equilibrium point x = 0 for the closed-loopsystem is asymptotic stable.
Corollary 1 results from the inverse optimality approach.
If the weights of the cost function are set to (33)–(35),the controller that satisfies inf u [Ṁ (x) + l(x, u)] = 0 is
analytically derived. The block diagram of the predictive
visual feedback control focused on the inverse optimality
approach is shown in Fig. 5.
IV. C ONCLUSIONSThis paper proposes a stabilizing receding horizon control for a 3D visual feedback system with fixed camera
configuration, which is a highly nonlinear and relatively
fast system, as a first step for a predictive visual feedback
control. It is shown that the stability of the receding
horizon control scheme is guaranteed by using the terminal
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