Proceedings of 2004 IEEE/RSJ International Conference onIntelligent Robots and Systems
September 28 - October 2, 2004, Sendai, Japan

Back Handspring Robot – Target
Dynamics-Based Control
Sang-Ho HyonNaoto Yokoyama

Takashi Emura

Dept. of Bioengineering & Robotics Dept. of Bioengineering. & Robotics
Dept. of Bioengineering & Robotics
Tohoku University
Tohoku University
Tohoku University
Sendai 980-8579, Japan
Sendai 980-8579, Japan
Sendai 980-8579, Japan
Email: sangho@ieee.org
Email: emura@emura.mech.tohoku.ac.jp

Abstract— This paper reports on a gymnastic robot, which
are developed for various floor exercises such as jumping,somersault and back handspring. The robot is a planarand serially connected four-link robot, whose joints are fully
actuated by electric servomotors. In this paper, the modelingand the controller for back handspring are addressed. The
controller is derived from task-specific target dynamics andits model matching. The use of global physical quantities suchas center of mass, or angular momentum allows even simple
target dynamics to generate complex gymnastic motions ofmulti-body system. The effectiveness of the controller isconfirmed via simulations and experiments.

I. I NTRODUCTION
Realization of complex and fast motion task is importantand attractive area of research. It enhances the mobilityof humanoid robots and also contributes to good understanding of motor control of human. Gymnastic motion
is included in such interesting motion tasks. This paper
reports on a gymnastic robot, which can perform jumping,somersault, and back handspring (Fig. 1).
There are some related studies on gymnastic robots.
Raibert and his co-workers succeeded in jumping andsomersault control of a 3D biped robot [1] [2]. Since
the robot has telescopic leg, jumping or flipping is rather
easy to realize by some intuitive controllers. However, it
becomes impossible to derive such a intuitive controllers
when the robot has articulated multi-link structure andhighly nonlinear dynamics. On the other hand, Arikawa
succeeded in jumping control of five-link robot by solving
numerically two-point boundary problem to obtain joint
trajectories and applying high-gain local feedback control
to each joint [3].
When human performs gymnastic motion, learning andits feed-forward control may plays dominant role. However,as long as the learning is based on sensory information,
feedback controllers can give important insights into thelearning mechanism, especially when the motion is generated only by feedback controller. Moreover, feedback
controllers have their own advantages over feed-forward
ones; a robustness against sudden change of environment is
the one of them. In this context, there are some successful
examples of high bar robot by using real-time feedback
controllers. Spong and Yamakita realized swing up andbalancing control [4][5]. Nakanishi succeeded in brachiating control of a two-link robot [6]. Recently Yamakita

0-7803-8463-6/04/$20.00 ©2004 IEEE

Fig. 1.

Four-link planar gymnastic robot

demonstrated continuous motion of swingup → giant →somersault → landing → balancing, by multi-link robot[7]. Nonlinear feedback control is also effective to jumping
and flipping motion. For example, Berkemeier applied
feedback linearization technique to underactuated robot(foot can freely rotate like a pivot joint) [8]. They foundzero-dynamics, which result from partial feedback linearization, to produce periodic hopping or flipping motion.
The same approach could be found in [9].
The objective of this paper is to propose a new control
framework of gymnastic robots that performs floor exercise. To do so, here we consider a full-actuated planar
model. The robot has four degrees of freedom (DOF),
which can describe “global dynamics" of planar gymnastic
motions as described later. From the theoretic point of view,
underactuated robots and their controllers are of interest.
However, to realize more complex and fast gymnastic
motions, we need “fully actuated" model. “Fully actuated"means the robot has a foot connected with actuated ankle
joint. Ankle torque can provide a large angular momentumof the total system around the contact point, which is248

TABLE I
R OBOT PARAMETERS
Mass [kg]Link length [m]*Position of C.M. [m]Inertia [kgmm2 ]

Link0

Link1

Link2

Link3

0.135
0.09
0.074
260

0.626
0.13
0.031
1200

0.533
0.15
0.55
880

0.579
0.19
0.086
1600

Z
Link 3(x ,z )gg

-ψ2

*Position of C.M. means the distance between C.M. of each link and theirproximal joint. All principal axes of each link are nearly coincident with
their center axes.

X

ψ3
Link 2
Joint 3
Link 1

Joint 2

ψ1
Joint 1

critical to gymnastic motion associated with high speed
rotation. It can also be used for balancing task.
Our control strategy is to impose some target dynamics
that describe gymnastic motions and realize them by model
matching controller. We expect the use of global physicalquantities such as center of mass, or angular momentum
allow even simple target dynamics to generate complex
gymnastic motions of multi-body system.

(x , z )0

Fig. 2.

0hLink 0ψ0Definition of mathematical model

II. ROBOT MODEL
A. Mechanical model development
Figure 1 shows a newly developed planar gymnastic
robot. The robot has four links, serially connected by three
joints. Each joint is actuated by geared servomotor (Maxon
20 W) through a timing belt. The total reduction ratio is10.8. Most of the mechanical parts are machined out ofaluminum frame. Overall height of the robot is 0.46 m andthe total weight is about 1.74 kg. Physical parameters are
summarized in Table 1.
The controller is implemented and runs in real time on
a Linux PC (PentiumIII 500MHz). The control period is
1.3 ms. Servomotors are driven by DC servo-drivers withthe torque control mode. Commanded signal to the servodriver is fed by DA converter and joint angles are obtainedby the encoders mounted to servomotors via digital I/O.
All the interface is provided by a ISA-bus interface board.
The robot does not carry the computer and servo-drivers
because it is at prototype stage. Also, gyro is not installed
because the motion is too fast. To detect the ground contactof the robot, touch switches are attached to the floor.

Stance I

Fig. 3.

Stance II
Touchdown II

Flight II

Lift-off II

Stance I
Touchdown I

Phase transition of back handspring

described by impulse equations. When human performs
back handspring, thanks to the hyper-multi-link structure,
the touchdown is very smooth and no “chattering" occurs
between the foots (or hands) and the floor.
We aim to such a smooth touchdown, but do not want tomake it complicated unnecessarily. That is why we introduce inelastic impulse assumption. To meet the assumption,a soft mat is covered on the floor.
C. Equations of motion at Stance I and Flight I
The equations of motion can be derived by Lagrange’s
formulation.
J(q)q̈ + H(q, q̇) = u + E T λ
E q̇ = 0,

B. Mathematical model representation
Figure 2 shows the definition of mathematical model.
To derive equations of motion for stance phase and flightphase at once, generalized coordinates includes absolute
position. That is, the generalized coordinates are definedas the position of the toe, (x 0 , z0 ), and the angles, ψ =(ψ0 , ψ1 , ψ2 , ψ3 ), where ψ0 is the attitude of the link 0(foot), and ψ 1 , ψ2 and ψ3 are the angles of joint 1 (ankle),
joint 2 (knee) and joint 3 (hip) respectively.
We assume the foot does not bounce back, nor slip
the ground (inelastic impulsive impact) This assumption isrequired to introduce impulse equation. As shown in Fig. 3,
a back handspring motion is composed of successive phasetransitions; Stance 1 (standing on the feet) → Flight 1→ Touchdown 2 → Stance 2 (standing on the head) →Touchdown 1 → · · ·. It is similar to the phase transitionof running motion. Motions of the touchdown phases are

Flight I

Lift-off I

(1)
(2)

where q = (x0 , z0 , ψ) ∈ R6 is the generalized coordinates,
J(q) ∈ R6×6 is a inertial matrix and H(q, q̇) ∈ R 6 isa nonlinear term. The nonlinear term includes joint-wiseviscous friction, which is identified by experiments. The
generalized force u ∈ R 6×1 contains joint torques τ =∂Φ0,1 (q)represents the(τ1 , τ2 , τ3 )T ∈ R3 . Matrix E :=∂qderivative of the constraint for the position of the toe andfor the posture the foot:


x0
(3)
Φ0 (q) =  z0  = 0,ψ0or, only for the position of the toe:x0
Φ1 (q) =
= 0.
z0

249

(4)

The associated constraint force is represented by λ 0 :=(λx , λz )T ∈ R2 or λ1 := (λx , λz , λψ )T ∈ R3 . They canbe calculated as:
λ = −(EJ −1 E T )−1 EJ −1 (u − H).

(5)

At the beginning, the robot keeps Eq. (3). If the zerocrossing of λψ is detected, the heel is off the ground andthe constraint switches to (4). Dynamics of flight phase canbe obtained by removing λ from (1).
We will further apply coordinate transformation so that
we can directly handle the position of the center of mass
(C.M.) of total system.
Using new generalized coordinates q g = (xg , zg , ψ),
Eq. (1) is transformed to:Jg (qg )q̈g + Hg (qg , q̇g ) = u + Eg (qg )T λg
Eg (qg )q̇g

= 0.

(6)

III. BACK HANDSPRING CONTROLLER
A. Target dynamics
Back handspring is a rapid and complex motion task
composed of multiple phases. In the control problem, it
is not realistic to depend on some pre-planned reference
trajectories, because in general such a trajectory-tracking
scheme cannot adapt to the changes of environment, which
are difficult to predict. Instead, we try to achieve complex
motion tasks by describing simple target dynamics (equation of motion) about global physical quantities such as
C.M. or momentum. The same approach could be foundin control of running robots (e.g. [11]).
1) Stance I
At Stance I with the heel supporting, target dynamics isdesigned as:
ẍ

(7)

M z̈
P˙

Note that (7) is a moving constraint because of the coordinate transformation (compare with (2)). Threfore, theconstraint force becomes:
λg = −(Eg Jg−1 EgT )−1 (γ + Eg Jg−1 (u − Hg )),(8)

where γ = ∂q∂g (Eg (qg )q˙g )q̇g .
Equation (6) has a decomposed structure of translationalmotion and rotary motion as shown below.
M ẍg
M z̈g

= λx
(9)
= −M g + λz
(10)


1
0
Jψ (ψ)ψ̈ + Hψ (ψ, ψ̇) =− R(ψ)λψ (11)τmHere, M = m1 + m2 + m3 is the total mass, g is gravity
acceleration, and R is a transformation matrix. J ψ ∈ R4×4
and Hψ ∈ R4 are the inertia matrix and the nonlinear terms
respectively, which are related only to ψ or ψ̇.
D. Equations of motion at Touchdown I
Touchdown equation can be expressed as Lagrange’s
impulsive equation, which can be found in some classicaldynamics textbook [10].



∆x˙g
M 0
02×4  
 0 M∆z˙g  = EgT λ̂,
(12)
04×2
Jψ
∆ψ̇
where ∆x˙g := ẋg+ − ẋg− , ∆z˙g := żg+ − żg− and ∆ψ̇ :=ψ̇+ − ψ̇− are the instantaneous velocity changes just before
and after touchdown, and λ̂ ∈ R2 is the constraint impulse
associated with the instantaneous velocity constraint attouchdown:ẋg+Eg  żg+  = 0.
(13)
ψ̇+Combining (12) and (13), we can calculate λ̂, ẋg+ , żg+ ,and ψ̇+ for given variables just before the impact.
Equations for Stance II, Flight II and Touchdown II canbe easily obtained by swapping coordinates.

g

= a

(14)

= −Kz (z − ze1 )(15)= −Kp (P g − Pgd ),(16)

where the “barred" variables implies those of target dynamics to be realized in Section 3.2. The first part is ahorizontal dynamics of constant acceleration, the second
part is simple oscillation around z e1 , and the last part isto control Pg , the angular momentum around C.M., to adesired value Pgd .
When the heel leaves the ground, the robot stands on
its toe. There is no available applied torque around contact
point: the system becomes to underactuated. In particular,
we cannot control P g arbitrarily in this phase. Therefore weabandon Eq. (16) by turning off the ankle torque, τ 1 = 0.
2) Flight I
At Flight I, there are no target dynamics designed,
for this time. Instead, joint angles are locally controlled
to a specified target configuration (ψ d = ψ1d , ψ2d , ψ3d )until the top of the link 4 (head) touches down. The
configuration is empirically determined. Here, touchdown
point is the most important because at the subsequent
phase, Stance II, there is no available applied torque aroundcontact point as the same as Stance I with toe supporting.
Touchdown point affects much on the rotational speed(horizontal speed) of the robot. We will pose this problem
to the future work and do not discuss here.
3) Stance II
At Stance II, target dynamics is set only to C.M.:
ẍ
M z̈

= a2
= −Kz (z − ze2 ).

(17)
(18)

4) Flight II
Controller at Flight II is the same as Flight I, exceptthat ψd is empirically determined to stop the motion of therobot.
B. Model matching control
This section derives control input τ to realize the above
target dynamics. First, remember Eq. (9) and (10). They saythat we can control ẍ g and z̈g arbitrarily via λx and λz ,250

which are nothing but ground reaction forces. Substituting(14) and (15) to (9) and (10), it seems we can solve τ from
Eq. (8).
Rearranging (8) yield the following equations.
Eg Jg−1 (u − Hg ) = γ − (Eg Jg EgT )λg
 


A1
B1



0
A11 A12− Hg = B1τ
A12 τ = B1 + AHg

(19)

(20)
(21)

Since A12 is non-invertible, to determine “uniquely" thecontrol inputs, we need another target dynamics. With thetarget dynamics about P g , we can solve it. Pg can be
expressed as:
Pg = J0 ψ̇0 + J1 ψ̇1 + J2 ψ̇2 + J3 ψ̇3 ,(22)

where Ji (i = 0, 1, 2, 3) is nonlinear inertia terms. Its time
derivative is calculated as:Ṗg = J0 ψ̈0 + J1 ψ̈1 + J2 ψ̈2 + J3 ψ̈3 + J4 .

(23)

Substituting (16), we get the following equation.
[ J0


J1

J2


J3 ]Jg−1 (u − Hg ) = P˙g


A2

B2

Combining (21) and (24) gives




A12
B1
τ=
+ AHg .
A2
B2

(24)

(25)

This time, the matrix of the left hand side is invertible andwe can obtain control input τ .
IV. S IMULATION AND EXPERIMENT
Parameters to be determined are;
• Initial configuration ψ(0)• a1 , Kz , ze1 , Pgd , a2 , ze2 (Stance I and Stance II)• Desired touchdown posture ψ d (Flight I and Flight II)
Moreover, we should consider about torque limits. In ourcase, it is about ±1.5 Nm.
These parameters are roughly tuned step by step along to
the following way. First ψ(0) is determined from x g (0) andzg (0). xg (0) is just above the ankle. Then, z g (0), a1 , Kz ,ze1 and Pgd in Stance I are tuned so that the head touches
down as fast as possible. Doing so, the flight time becomesshort. Therefore, we can choose ψ d as the terminal positionof Stance I. If the angular momentum P g has enough level,
it is easy for the robot to rotate around its pivot (head) in
Stance II. We first try to set Pgd to some constant value tobe reached, but the results are not satisfactory; the robot
rotates too fast than the translational motion of (x g , zg ).
Therefore, we modify it byPgd = −c(xg − xg (0)),

II: Pg is not controlled in this phase. This implies wecannot expect enough P g at the second lift-off. Therefore,
we set ze2 and Kz2 larger than those of Stance I to makethe robot take off higher. The higher the robot jumps, themore flight time for the controlling final configuration is
given, even if P g is not enough. As in Stance I, admissible
region of these parameters is rather limited because theystrongly affect on the posture of the robot. The final
parameter ψd in Flight II is tuned so that the robot canstop and standstill. In simulation we can control the robot
to standstill in the next Stance I, but in experiment the
robot is controlled to have some final posture, which isunchanged until it stops.
One of the parameter set thus determined is shown in Table 2. Using these parameters, successful back handspring
motion is performed in both simulation and experiment.
Fig. 4 depicts the animation created by simulation. Snap
shots of corresponding experiment are shown in Fig. 5.
Since the robot is not installed with high-speed gyro (itrequires more than 1000 deg/s !), in the experiment the
robot is controlled in open-loop. The video of the experiment is included in the conference proceedings. Fig. 6 andFig. 7 show the joint angles and joint torques, together
with ON/OFF state of floor switches (On = 1, Off = 0)
at the simulation and experiment. Each joint torque in theexperiment is obtained by motor current via AD converter.
Although the back handspring finishes within a very
short period of time (about 0.6 sec!), these two figures
show clearly the time evolution of each variables andphase transitions. Until the first touchdown (0.3 s), there
are no significant differences between the simulation andexperiment: model matching controller works perfectly.
After the touchdown, however, the differences becomelarger. The most significant difference is the period of
Stance II. The stance period of simulation is about 0.1
s, while that of experiment is about 0.05 s. This implies
the difference mainly comes from the modeling error oftouchdown phase: we can say the mat covers the floor
does not emulate inelastic impulse assumption well. But weare optimistic about this error because we can take otheroptions (e.g. installing dampers to the top of the robot).
V. C ONCLUSION
We developed a gymnastic robot for various floor exercise such as jumping, somersault and back handspring.
The robot is a planar and serially connected four-link robot,
whose joints are actuated by electric servomotors. Then we
described about our control strategy of fast and complex
gymnastic motion, which is different from the classical
TABLE II

(26)

C ONTROL PARAMETERS
Parameterψ1 (0), ψ2 (0), ψ3 (0)a1 , ze1 , Kz , c @ Stance I
a2 , ze2 , Kz @ Stance II
ψ1d , ψ2d , ψ3d @ Flight I
ψ1d , ψ2d , ψ3d @ Flight II

where c is a constant. This comes from our observation thatgymnast increases his rotational speed when C.M. proceeds
enough.
The parameters in Stance II are difficult to choose. The
difficulty comes from the underactuated structure of Stance

251

Value
120, -94, 90
4.5, 0.19, 20, 4.2
3.8, 0.56, 40
0, -70, -100
125, -125, 50

Fig. 4.

Fig. 5.

Stick animation of back handspring simulation

Snap shots of back handspring experiment252ψ1 [deg]

200
100

exp.
sim.

ψ2 [deg]

0
0
0

0.1

0.2

0.3

0.4

−100
−200
0
200

ψ3 [deg]

0.5

0.6

0.5

0.6

exp.
sim.

0.1

0.2

0.3

0.4

0

exp.
sim.

switch

−200
0
1

0.1

0.2

0.3

0.4

0.5

0.6

exp.
sim.

0.5
0
0

0.1

0.2

0.3
Time [s]

0.4

0.5

0.6

This is known as aerial attitude control problem with nonholonomic constraint, discussed in [12] [13] [14] [15].
Solution of this problem will leads to the systematic designof the control parameters of target dynamics at stance phaseas well.
In this context, re-design of controlled variables is also
important task because the robot has three control inputs,
hence the maximum number of the target dynamics that canbe followed independently is three. For example, insteadof the position of x g , we can introduce the target dynamics
about configuration of the robot because the configuration
at lift-off strongly affects on the aerial motion.
We believe more difficult task such as successive backhandspring can be realized in the near future.
ACKNOWLEDGEMENT

Fig. 6.

τ1 [Nm]20

τ2 [Nm]R EFERENCES

0.1

0.2

0.3

0.4

0.5

0.6
exp.
sim.

0

−2
0
2
τ3 [Nm]The authors would give thanks to M. Suzuki for his great
help in fabricating the hardware.

exp.
sim.

−2
0
2

0.1

0.2

0.3

0

0.4

0.5

0.6

0.5

0.6

exp.
sim.

−2
0
1
switchJoint angles in simulation and experiment

0.1

0.2

0.3

0.4

exp.
sim.

0.5
0
0

0.1

Fig. 7.

0.2

0.3
Time [s]

0.4

0.5

0.6

Joint torques in simulation and experimentcontrol method that requires pre-planned reference trajectories. It was composed of task-specific target dynamics andits model matching. The use of global physical quantities
such as center of mass, or angular momentum allowed evensimple target dynamics to generate complex gymnastic
motions of multi-body system. The effectiveness of thecontroller was confirmed via simulations and experimentsof back handspring.
To the best of our knowledge, it is the first time thatback handspring is performed by real multi-link robot. In
simulation, we have also succeed in other interesting motion such as hopping, somersault. The somersault control,
however, needs more joint torques than back handspringand not applicable to the current machine.
Since the model matching controller is working well, theremained important task is a systematic design of target
dynamics. We think the key is the analysis and control ofintermediate flight phase, which are not provided in thispaper. If the lift-off configuration lies in some accessible
region and there are enough flight time, we can steer the
posture to some desired region until the next touchdown.

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