Advanced Robotics, Vol. 20, No. 1, pp. 93– 113 (2006)
 VSP and Robotics Society of Japan 2006.

Also available online - www.vsppub.com

Full paper
Back handspring of a multi-link gymnastic robot —reference model approach
SANG-HO HYON 1,∗ , NAOTO YOKOYAMA 2 and TAKASHI EMURA 1
1 Department of Bioengineering and Robotics, Graduate School of Engineering, Tohoku University,

Aramaki-Aza-Aoba 01, Aoba-ku, Sendai 980-8579, Japan

2 Makino Corp., Nakane 2-3-19, Meguro-ku, Tokyo 152-8578, Japan

Received 5 November 2004; accepted 14 February 2005
Abstract—This paper reports on the first gymnastic robot that can perform back handspring. The
robot is a planar and serially connected four-link robot, with its joints actuated by electric servomotors.
The paper describes the modeling of the robot and the control framework for a back handspring. The
controller is derived from a task-specific reference model and its model matching. The use of areference model described by global physical quantities such as center of mass or angular momentum
allows the gymnastic motion planning of a multi-body system to be intuitive and the model matching
controller can be applied directly to the experimental model without obtaining each joint trajectory.
The controller effectiveness is confirmed via simulations and experiments of the back handspring.
Although there remains the problem of how to systematically design the control parameters, the paper
shows the strength of the model-based controller for fast gymnastic motions.
Keywords: Gymnastic robot; back handspring; jumping; non-linear control; reference model.

1. INTRODUCTION

Realization of complex and fast motion tasks is an important and attractive areaof research. Such motion tasks enhance the mobility of humanoid robots and also
contribute to the better understanding of human motor control. Gymnastic motion
is included in such interesting motion tasks. This paper describes a gymnastic robotthat can perform jumps, somersaults and back handsprings (Fig. 1).
Some studies have addressed gymnastic robots. Raibert et al. succeeded injumping and somersault control of a three-dimensional bipedal robot [1, 2]. In thesomersault, the robot undertakes rotational motion in flight, as well as translationalmotion (jumping). How much the robot rotates in the air depends on the angularmomentum around the center of mass (CM) at lift-off and the inertia around the∗ To whom correspondence should be addressed. E-mail: sangho@ieee.org

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Figure 1. Four-link planar gymnastic robot.

CM during the flight. Analysis of the dynamics shows that we can control the initial
angular momentum by the extension force of the leg for a given distance between
the hip joint and the CM of the body. Furthermore, we can control the rotation
speed by extending or retracting the legs in mid-flight. We depend greatly on model
dynamics to obtain complex gymnastic performances for an articulated and multilink structured model.
However, there is also an example in which jumping of the five-link robot
was performed by numerically solving a two-point boundary problem to obtainjoint trajectories and then applying high-gain local feedback control to eachjoint [3].
When humans perform gymnastic motions, feed-forward control with learning
may play a dominant role. However, as long as that learning is based on sensory
information, feedback controllers can give important insights into the learning
mechanism, especially when the motion is generated only by a feedback controller.
Moreover, feedback controllers have their own advantages over feed-forward
ones — one of them being robustness against sudden environmental change.
In this context, there are some successful examples of a high bar robot using
real-time feedback controllers. Spong and Yamakita et al. realized swing-up andbalancing control [4, 5]. Nakanishi et al. succeeded in brachiating control of a twolink robot [6]. Recently, Yamakita et al. demonstrated a continuous motion of swing
up → giant → somersault → landing → balancing using a multi-link robot [7].
Non-linear feedback control is also effective for jumping and flipping motions.

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For example, Berkemeier and Fearing applied the feedback linearization techniqueto an underactuated robot (whose foot can freely rotate as a pivot joint) [8].
They found zero dynamics, which result from partial feedback linearization, to
produce a periodic hopping or flipping motion. The same approach was foundin [9].
This paper is intended to propose a control framework and a robotic test bed to
study skill control in gymnastic motions [10]. For that purpose, a planar robot model
is considered. The robot has 3 d.o.f., which can describe the ‘global dynamics’ ofplanar gymnastic motions, as described later.
The control strategy in this paper is to impose some reference model that describes
gymnastic motions, especially for floor exercises, and realize them by a model
matching controller. We expect the use of global physical quantities such as the CM
or angular momentum to allow even a simple reference model to generate complex
gymnastic motions of a multi-body system.

2. ROBOT MODEL2.1. Mechanical model development
Figure 1 shows the newly developed planar gymnastic robot. The robot has four
links that are connected serially by three joints. Each joint is actuated by a geared
servomotor (20 W; Maxon Precision Motors) through a timing belt. The total
reduction ratio for each joint is 10.8. Most of the mechanical parts are machined outof an aluminum frame. The overall height of the robot is 0.46 m; its total weight is
1.74 kg. Table 1 summarizes the physical parameters. Note that there is no ‘boom’to constrain the robot to the sagittal plane.
Figure 2 shows the computer and electric system. The controller can run inreal-time on a Linux PC (Pentium III 500 MHz) with a 1.3-ms control period.
Servomotors are driven by DC servo-drivers with the torque control mode. The
command signal to the servo-driver is sent by the DA converter. Joint angles are
obtained by encoders mounted to the servomotors via a digital I/O. The interface
is provided by an ISA-bus interface board (RIF-01). The robot is connected to theTable 1.
Link parameters

Mass (kg)
Link length (m)CM position (m)aInertia (kg mm2 )

No. (i)

1

2

3

4

mi
li
lci
Ji

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

a The CM position is the distance between the CM of each link and their proximal joint.

All
principal axes of each link are nearly coincident with their center axes. The ankle height h is 0.03 m.

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Figure 2. Computer and electric system.

computer and servo-drivers with cables. We use very light cables so that they do not
disturb the robot’s motion. Moreover, a gyro is not installed because the bandwith
is not satisfactory for our purpose. Touch switches are attached to the floor to detect
the ground contact of the robot.
2.2. Mathematical model representation
Figure 3 shows the definition of the mathematical model. Generalized coordinates
include the absolute position to derive equations of motion for the stance phase andflight phase at once. That is, the generalized coordinates comprise (x0 , z0 ), the toe
position (xg , zg ), the position of CM, as well as the joint angles ψi (i = 1, 2, 3, 4),
where ψ1 is the attitude of the link 1 (foot), and ψ2 , ψ3 and ψ4 are the angles of theankle, knee and hip, respectively. Control inputs are the joint torques, τ2 , τ3 and τ4 .
There is no torque applied around the toe (τ1 = 0).
We assume that the foot neither bounces back nor slips on the ground (inelastic
impulsive impact). This assumption is required to introduce an impulse equation.
As shown in Fig. 4, a back handspring motion comprises successive phase transitions: Stance I (standing on the feet) → Flight II → Touchdown II → Stance II(standing on the head) → Touchdown I → · · ·. It is similar to the phase transition ofa running motion. Impulse equations describe the motions of the touchdown phases.
When a human performs a back handspring, thanks to their hyper-multi-link struc-

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Figure 3. Definition of the mathematical model.

Figure 4. Phase transition of the back handspring.

ture, the touchdown is very smooth and no ‘chattering’ occurs between the feet (orhands) and the floor.
We aim to achieve such a smooth touchdown, but do not want to make it unnecessarily complicated. Therefore, we introduce an inelastic impulse assumption. A soft
mat covers the floor to provide for that assumption.
2.3. Equations of motion at Stance I
The equations of motion can be expressed by Lagrange’s equation of motion withconstraints. A CM-coordinate system is used so that the CM position appears
explicitly in the mathematical representation. Using the generalized coordinates

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qg = [xg , zg , ψ]T ∈ R 6 , where ψ := [ψ1 , ψ2 , ψ3 , ψ4 ]T ∈ R 4 , the equation of
motion can be expressed as:
Jg (qg )q̈g + Hg (qg , q̇g ) + Gg (qg ) = u + Eg (qg )T λg ,Eg (qg )q̇g = 0.

(1)
(2)

Details of the inertial matrix Jg (qg ) ∈ R 6×6 , non-linear term Hg (qg , q̇g ) ∈ R 6 andgravity term Gg (qg ) ∈ R 6 are provided in the Appendix. The generalized forceu ∈ R 6 is represented as:


0
(3)
u = 3×1 ,τwhich includes τ = [τ2 , τ3 , τ4 ]T ∈ R 3 . The matrix Eg (qg ) represents the derivativeof the constraints as shown below.
As the constraint of the heel contact is defined as:
 
x0
(4)
0 (q) =  z0  = 0,
ψ1
the time differentiation of the equation:


1
d
I
− J
0 (q) =  2×2 M 12  q̇g = Eg (qg )q̇g = 0,
dt
1000
0

(5)

1×2

yields (2), where M = m1 + m2 + m3 + m4 is the total mass and the matrix
J12 ∈ R 2×4 is defined in the Appendix. In this case, Eg (qg ) ∈ R 3×6 .
However, for toe contact, the constraint becomes: x1 (q) = 0 = 0.
(6)
z0
Its time differentiation becomes:d1
1 (q) = I2×2 − J12 q̇g = Eg (qg )q̇g = 0.
dt
M

(7)

In this case, Eg (qg ) ∈ R 2×6 .
The associated constraint forces (the ground reaction forces) are represented by
λg := (λx , λz , λψ )T ∈ R 3 for heel contact or λg := (λx , λz )T ∈ R 2 for toe contact.
They can be calculated by combining (1) and (2), which yields:(8)
λg = −(Eg Jg−1 EgT )−1 γ + Eg Jg−1 (u − Hg − Gg ) ,where γ =∂(Eg (qg )q˙g )q̇g .
∂qg

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At the beginning of the motion the robot stands still and maintains the constraint
condition (4). Then, if the zero crossing of λψ is detected, the heel takes off theground and the constraint switches to (6).
From (A.9) and (A.10) in the Appendix, (1) is shown to have a decomposed
structure of translational motion and rotary motion, as shown below:  M 0x¨g00(2×4)      z¨g  =  −Mg  0 M
Hψ (ψ, ψ̇)ψ̈
0(4×2) Jψ (ψ)



02×1
 
0

1  

 λx
 


0
0  λ  + 
+
 ,
z
 − 1 J12

0
 
 M

0  λψ

τ
I2×2

(9)

0
where g is the acceleration due to gravity. The matrix Jψ ∈ R 4×4 and the vectorHψ ∈ R 4 are shown in the Appendix. Note that λψ in the second term in the righthand side of (9) becomes zero in the case of toe contact.
2.4. Equations of motion at Flight I
The dynamics of the flight phase can be obtained easily by removing λg from (1),which yields:
Jg (qg )q̈g + Hg (qg , q̇g ) + Gg (qg ) = u.

(10)

2.5. Equations of motion at Touchdown I
The touchdown equation can be expressed as Lagrange’s impulsive equation, which
can be found in some classical dynamics textbooks [11]. The change of thegeneralized velocity just before and after touchdown q̇ := q̇+ − q̇− can becalculated by the impulse equation with the constraint as
Jg (qg )q̇g = EgT λ̂g ,Eg q̇g+ = 0,

(11)
(12)

where Eg is the same matrix in (7) and λ̂g ∈ R 2 is the associated impulsive force,
which can be calculated as:
λ̂g = −(Eg Jg−1 EgT )−1 Eg q̇g− .

(13)Note that the touchdown contact occurs almost always at the toe or the head; hence,
only 1 (q) is considered.

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As in the case of Stance I, the impulse equation has a decomposed structure:



M 0
x˙g
02×4    z˙g 
 0 M
ψ̇
04×2 Jψ
02×1

1   λx0 λ .

= 1
z

J
−
12

 M
0  λψ



I2×2

(14)

0
2.6. Equations of motion at Stance II, Flight II and Touchdown II
Equations for Stance II, Flight II and Touchdown II can be obtained easily byswapping the angle coordinates (ψ1 , ψ2 , ψ3 , ψ4 ) ←→ (ψ4 , ψ3 , ψ2 , ψ1 ) in theequations.
3. BACK HANDSPRING CONTROLLER

3.1. Reference model
The back handspring is a rapid and complex motion task comprising multiple
phases. In the control problem, it is unrealistic to depend on some pre-planned
reference trajectories because, in general, such a trajectory-tracking scheme cannot
adapt to environmental changes, which are difficult to predict. Instead, we attempt
to achieve complex motion tasks by describing a simple reference model (equationof motion) about global physical quantities, such as the CM position or angularmomentum around the CM.
Our method describes a reference model of hopping, somersaults and handsprings
using common ‘spring dynamics’ in the ‘stance phase’. Spring-like behavior of
running is well known [12]; some examples exist in which spring-like behavior
is imposed for controlling running robots [13]. Spring-like dynamics are usefulwhen humans need to preserve mechanical energy during dynamic motion because
they have tendons in their joints. In a hopping or running motion, humans canutilize their joint tendons to store the energy, which engenders reduction of power
consumption. With appropriate parameters, tendon-driven running or hopping will
generate spring-like behavior [1, 12]. In other words, spring-like dynamics are well
suited to a tendon-driven mechanism. Discussions about how to introduce tendons
into the articulated model are beyond the scope of this paper (but see Ref. [14], forexample).
However, there are reasons why this paper gives a refrence model only to thestance phase. First, some floor exercises, such as the back handspring, do not haveBack handspring of a multi-link gymnastic robot

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sufficient flight time. When the flight period is short, the robot behavior in theflight phase is almost determined by the terminal state of the stance phase. Second,
the model matching condition for spring dynamics is met only at the stance phase.
There is no ground reaction force available in the flight phase and the robot cannot
control the CM.
If sufficient flight time is available, it becomes important to control the terminal
state of the robot within a given flight time. If the lift-off configuration lies insome accessible region for a given flight time, the posture can be steered to somedesired region until the next touchdown. This problem leads to non-holonomic
attitude control. A long-jump or somersault involves this kind of attitude control
in the aerial phase. Basically, the posture can be controlled by actively controllingthe moment of inertia around the CM [2]. Derivation of the controller, as well as
the identification of its accessible region, is rather easy for an articulated two-link
model [15]. However, for the three or more linked model, it becomes more difficult
to analyze the model and derive controllers [16, 17]. This paper does not discuss
the non-holonomic controller; it will be introduced in future work, where otherinteresting gymnastic motions with longer flight time, such as somersaults, will be
realized by the robot.
Below, we define the reference model for the back handspring.
3.1.1. Stance I.
designed as:

At Stance I with the heel supporting, the reference models are
M ẍ g = a1 ,(15)

M z̈g = −Kz (zg − ze1 ),P˙ = −K (P − P ),

(16)

g

p

g

gd

(17)

where the ‘barred’ variables imply those of reference model dynamics to befollowed by the controller in Section 3.2. The first two parts define spring-likedynamics of the CM, where a is the acceleration and ze1 is the virtual equilibrium.
The last equation is the exponential convergence of Pg , the angular momentum
around CM, to a desired value Pgd .
When the heel leaves the ground, the robot stands on its toe. There is no applied
torque available around the contact point. In particular, we cannot control Pg
arbitrarily in this phase because of the relationship:d(18)
Pg = Mgr sin θ,
dt
where r is the distance between the CM and toe, and θ is the angle. Therefore, (17)is abandoned and the reference model in this phase becomesM ẍ g = a1 ,(19)

M z̈g = −Kz (zg − ze1 ).

(20)

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3.1.2. Stance II. As in the case of toe contact in Stance I, reference models are
set only to CM:
M ẍ g = a2 ,(21)

M z̈g = −Kz (zg − ze2 ).

(22)

3.1.3. Flight I and Flight II. There are no reference models designed for theflight phase in the back handspring. Instead, joint angles are controlled locally to a
specified target configuration (ψd = ψ2d , ψ3d , ψ4d ) until the top of the link 4 (head)
touches down.
3.2. Model matching control
This section derives model matching control input τ to realize the above reference
model. Using this controller, the robot becomes dynamically equivalent to thereference model.
Substituting (15) and (16) into (8) yields:
Eg Jg−1 (u − Hg − Gg )

A




a1



= −γ − Eg Jg EgT  −Kz (zg − ze1 ) + Mg ,
λψ


(23)

B

or:


A1 A2







03×1
− Hg − Gg = B,τ(24)

where A1 ∈ R 3×3 , A2 ∈ R 3×3 and B ∈ R 3 . Rearranging this equation gives:
A2 τ = B + A(Hg + Gg ).

(25)

Therefore, if we specify λψ in (23), the control input τ can be uniquely determined.
It is shown below that λψ is related to the angular momentum. With the kinetic
energy K = 12 q̇gT Jg q̇g , the angular momentum Pg can be obtained by:
Pg =

∂K
,
∂ψ1

(26)

because ψ1 is the cyclic variable [11]. This equation results in:
Pg = Jg33 ψ̇1 + Jg34 ψ̇2 + Jg35 ψ̇3 + Jg36 ψ̇4 ,(27)

where Jg3i (i = 3, 4, 5, 6) shows the elements of the third row of Jg (see Appendix).
Its time derivative is calculated as:Ṗg = Jg33 ψ̈1 + Jg34 ψ̈2 + Jg35 ψ̈3 + Jg36 ψ̈4 − Hψ1 (qg , q̇g ),(28)

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where Hψ1 (qg , q̇g ) is the first element of Hψ in (9). Substituting the reference model(17) into this equation yields:[ Jg33 Jg34 Jg35 Jg36 ]ψ̈ = −Kp (P g − Pgd ) + Hψ1 .

(29)

On the other hand, the lower part of (9) gives: 1
λ[ Jg33 Jg34 Jg35 Jg36 ]ψ̈ = Hψ1 − J12 x + λψ .
λz
M

(30)

Thus, combining (29) with (30), then substituting (15) and (16), we can obtain therelationship between λψ and the reference model about Pg :1a1
− Kp (P g − Pgd ).
(31)
λψ = J12−Kz (zg − ze1 ) + Mg
MHowever, in the case of toe contact in Stance I, or Stance II, λψ dissappears in(23), and the dimensions of A and B shrink; A1 ∈ R 2×3 , A2 ∈ R 2×3 and B ∈ R 2 . In
this case, although the model matching condition still holds, control inputs cannot
be chosen ‘uniquely’ because A2 is non-invertible. Therefore, the pseudo-inverse
A#2 = AT2 (A2 AT2 )−1 is introduced.

4. SIMULATION AND EXPERIMENTThe parameters to be determined are:
• Initial configuration ψ(0)• a1 , Kz , ze1 , Pgd , a2 , ze2 (Stance I and Stance II)• Desired touchdown posture ψd (Flight I and Flight II).
Moreover, we should consider the joint torque limit of ±1.5 Nm.
These parameters are roughly tuned at this first stage. A reasonable way to find theparameters is an ongoing task (see Section 5). However, for reference, we describe
how we get feasible parameters for back handsprings step by step. First, ψ(0) is
determined from xg (0) and zg (0), where xg (0) is set to the same horizontal positionof the ankle joint. Then, zg (0), a1 , Kz , ze1 and Pgd in Stance I are tuned so thatthe robot touches down on the ground as soon as possible, thereby shortening theflight time. For that reason, we can choose ψd as the terminal position of Stance I. If
angular momentum Pg has a sufficiently high 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 unsatisfactory; the robot rotates much more quickly
than the translational motion of (xg , zg ). Therefore, we modify it by:(32)
Pgd = −c xg − xg (0) ,where c is a constant. This comes from our observation that a gymnast increases his
rotational speed when the horizontal position of the CM proceeds sufficiently.

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The parameters in Stance II are difficult to choose. The difficulty arises fromthe underactuated structure of Stance II: Pg can not be controlled arbitrarily in thisphase, as described in A.1). This fact implies that we cannot expect enough Pg atthe second lift-off (if the robot has arms with shoulder joints, the controllability will
be greatly increased. This will be done in our next stage). Therefore, we set ze2 andKz2 larger than those of Stance I to make the robot take off higher. The higher the
robot jumps, the more flight time is available for controlling the final configuration,even if Pg is not sufficient. The final parameter ψd in Flight II is tuned so that the
robot can stop and stand still.
Table 2 shows one parameter that is set in that manner. Using these parameters,a successful back handspring motion is performed in both the simulation andexperiment. Figures 5 and 6 show the time evolution of each joint angle and joint
torque, together with the On/Off state of the ground contact (On = 1, Off = 0) at
the simulation and experiment. In this experiment, the On/Off state is determinedby microswitches installed under the mat; each joint torque is obtained by motor
current via an AD converter. The robot does not contain a high-speed gyro (morethan 1000◦ /s). Therefore, the commanded torque is determined by simulation,
except for the PD-feedback in the two flight phases and the final phase (stopping
the motion).
Figure 7 depicts an animation created by simulation; Fig. 8 shows snap shots ofthe corresponding experiment, where the sequence of Touchdown I, Lift-off I and
Touchdown II is captured clearly.
Although the back handspring finishes within a very short period of time (around
0.6 s), the behavior of the robot, as well as phase transitions, are captured clearlyin the graphs and videos. From 0.0 to 0.2 s, the robot contacts the ground with its
feet. From 0.2 to 0.3 s, the robot contacts the ground with its toes. Figure 6 showsthat the joint torques are saturated in this phase, regardless of the ‘optimal’ torquedistribution by pseudo-inverse. This demonstrates the difficulty of controlling theCM without a foot: the robot requires maximum torque to follow the reference
model.
Until the first touchdown (0.3 s), there are no marked differences between thesimulation and experiment: the model identification is almost perfect. However, thedifferences become larger after the touchdown. The most remarkable difference isTable 2.
Control parameters
Parameter

Unit

Value

ψ2 (0), ψ3 (0), ψ4 (0)a1 , ze1 , Kz , c at Stance I
a2 , ze2 , Kz at Stance II
ψ1d , ψ2d , ψ3d at Flight I
ψ1d , ψ2d , ψ3d at Flight II

deg
—
—
deg
deg

120, −94, 90
4.5, 0.19, 20, 4.2
3.8, 0.56, 40
0, −70, −100
133, −129, 120

Back handspring of a multi-link gymnastic robot

Figure 5. Joint angles in the simulation and experiment.

Figure 6. Joint torques in the simulation and experiment.

105

Figure 7. Stick animation of the back handspring simulation corresponds to the data in Figs 5 and 6.

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Figure 8. Snap shots of the back handspring experiment corresponding to data in Figs 5 and 6; the index numbers correspond to those of Fig. 7, but the timestamps do not coincide after no. 4.

Back handspring of a multi-link gymnastic robot
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the period of Stance II. The stance period of the simulation is about 0.1 s, whereasthat of the experiment is about 0.05 s. This fact implies that the difference mainly
arises from the modeling error of the touchdown phase; apparently, the mat covering
the floor does not emulate the inelastic impulse assumption well. The stance period
affects the spring dynamics, especially for vertical motion. If the period is too short,the vertical speed of the CM cannot be restored. For this reason, the ground levelat the final landing is slightly lower than the starting level. However, the modelingerror can be overcome by installing hands, by which spring dynamics should be
followed at Stance II, as in Stance I. Although the experiment can be successful
only by such on-the-spot tuning, the repeatability of the experiment is very highexcept for its termination (we do not use active balancing control).

5. CONCLUSION

This paper proposed a control framework and a robotic test bed to study skill
control in gymnastic motions by presenting a specific example — a back handspring
motion. For this purpose, a planar and serially connected four-link robot wasdeveloped. Its joints are actuated by electric servomotors. Then a control frameworkfor fast and complex gymnastic motion was proposed. It comprised a task-specificreference model and model matching control; it differs from the classical control
method that requires pre-planned reference trajectories. The use of global physicalquantities such as the CM or angular momentum allowed even a simple reference
model to generate complex gymnastic motions of a multi-body system. The
effectiveness of the controller was confirmed via simulations and experiments using
a back handspring motion. This is the first time that a back handspring has been
performed by a real multi-link robot.
However, this is our first stage. Two main tasks remain for future work:(i) Touchdown control to give appropriate initial angular momentum for the next
stance phase.
(ii) Systematic design of the control parameter or re-design of the reference model.
The first task will lead to stability of the floor exercise. As stated in Section 2,
reference models in the air are not provided in this paper because they are
not necessary for the back handspring motion. Thereby, we have succeeded inperforming a back handspring without active posture control at the flight phase. The
only exception will be control of the touchdown position of the foot (or hand). The
touchdown position governs the dominant rotational behavior at the stance phasebecause the torque around the toe is not available; alternatively, the torque of theankle joint (or wrist joint) is limited by the zero-moment point. Controlling thetouchdown position leads to ‘gait stability’ of the consecutive handspring motion orit helps to terminate the gymnastic motion. Instead of controlling all states using
a non-holonomic attitude controller, it will be rather realistic to give priority to thetouchdown position. For those reasons, derivation of the controller is our next step.

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The second task poses the most important problem because it includes the key toskill control. A general solution to determine control parameters such as virtual
spring constant or desired angular momentum according to specific gymnastic
motions is needed. In this context, re-design of the reference model should also
be considered as an alternative.
After solving these two problems, a non-holonomic attitude controller at the flightphase will be combined effectively with the stance controller and used for the
gymnastic motions with a longer flight time.
Acknowledgements
This study was partially supported by the Ministry of Education, Science, Sports
and Culture, Grant-in-Aid for Young Scientists (B), 15760144, 2003. The authors
thank Masatoshi Suzuki for his help in fabricating the hardware.

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APPENDIX: DERIVATION OF EQUATION OF MOTION WITHOUTCONSTRAINT

The appendix provides details of equations of motion without constraints. These
are the flight phase dynamics. Presuming the kinetic energy of the robot as
K = (1/2)q̇ T J (q)q̇, the coordinate transformation from q = [x0 , z0 , ψ]T to
qg = [xg , zg , ψ]T is performed as:
1
1
1
K = q̇ T J (q)q̇ = q˙g R(ψ)T J (q)R(ψ)q˙g = q˙g Jg (ψ)q˙g ,
2
2
2

(A.1)

∂q
is the transformation matrix. Since the CM position can bewhere R(ψ) := ∂qgcalculated as:   
x0
xg
=
zg
z0

−

1
M




b1 C1 + b2 C12 + b3 C123 + b4 C1234,
b1 S1 + b2 S12 + b3 S123 + b4 S1234(A.2)

where:
M = m1 + m2 + m3 + m4
b1 = −m1 lc1 − (m2 + m3 + m4 )l1
b2 = −m2 lc2 − (m3 + m4 )l2
b3 = −m3 lc3 − m4 l3
b4 = −m4 lc4S12··· = sin(ψ1 + ψ2 + · · ·)C12··· = cos(ψ1 + ψ2 + · · ·)the transformation matrix R(ψ) can be represented as:


1
1 0
R(ψ) =− J∈ R 2×6 ,0 1 M 12
where:

−b1 S1
J12 =b1 C1−b2 S12b2 C12−b3 S123
b3 C123−b4 S1234
b4 C1234(A.3)
∈ R 2×4 ,is the submatrix of J (q), the inertia matrix of the toe coordinate system.

(A.4)

Back handspring of a multi-link gymnastic robot

Inertial matrix Jg (ψ) in (A.1) is calculated as the following.


M 0 0
0
0
0
0 M 0000  0 0 Jg33 Jg34 Jg35 Jg36  ∈ R 6×6 ,Jg (ψ) =  0 0 Jg34 Jg44 Jg45 Jg46 
 0 0 Jg35 Jg45 Jg55 Jg56 
0 0 Jg36 Jg46 Jg56 Jg66
Jg33 = 2(k5 C2 + k6 C3 + k7 C4 + k8 C23 + k9 C34 + k10 C234
+ k11 S2 + k12 S23 + k13 S234 ) + k1 + k2 + k3 + k4 + k14 ,Jg34 = k5 C2 + 2k6 C3 + 2k7 C4 + k8 C23 + 2k9 C34 + k10 C234
+ k11 S2 + k12 S23 + k13 S234 + k4 + k2 + k3 ,Jg35 = k6 C3 + 2k7 C4 + k8 C23 + k9 C34 + k10 C234
+ k12 S23 + k13 S234 + k4 + k3 ,Jg36 = k7 C4 + k9 C34 + k10 C234 + k13 S234 + k4 ,Jg44 = 2k6 C3 + 2k7 C4 + 2k9 C34 + k2 + k3 + k4 ,Jg45 = k6 C3 + 2k7 C4 + k9 C34 + k3 + k4 ,Jg46 = k7 C4 + k9 C34 + k4 ,Jg55 = 2k7 C4 + k3 + k4 ,Jg56 = k7 C4 + k4 ,Jg66 = k4 ,k1 = (Ma11 − b12 )/M,
k2 = (Ma22 − b22 )/M,
k3 = (Ma33 − b32 )/M,
k4 = (Ma44 − b42 )/M,
k5 = (Ma12 − b5 b2 )/M,
k6 = (Ma14 − b2 b3 )/M,
k7 = (Ma23 − b4 b3 )/M,
k8 = (Ma13 − b5 b3 )/M,
k9 = (Ma45 − b2 b4 )/M,
k10 = (Ma24 − b5 b4 )/M,
k11 = (Ma25 + b1 b2 )/M,
k12 = (Ma34 + b1 b3 )/M,
k13 = (Ma35 + b1 b4 )/M,
k14 = (Mah − b52 )/M,
b5 = −(m2 + m3 + m4 )h,

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(A.5)

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S.-H. Hyon et al.

a12 = (m2 lc2 + m3 l2 + m4 l2 )h,a13 = (m3 lc3 + m4 l3 )h,a14 = (m3 lc3 + m4 l3 )l2 ,a23 = m4 lc4 l3 ,a24 = m4 lc4 h,
a25 = −(m2 lc2 + m3 l2 + m4 l2 )l1 ,a34 = −(m3 lc3 + m4 l3 )l1 ,a35 = −m4 lc4 l1 ,2
+ (m2 + m3 + m4 )l12 ,a11 = J1 + m1 lc1
2
a22 = J2 + m2 lc2+ (m3 + m4 )l22 ,2+ m4 l32 ,a33 = J3 + m3 lc32,
a44 = J4 + m4 lc4
a45 = m4 lc4 l2 ,ah = (m4 + m3 + m2 )h2 .
Then, with the new CM-coordinate system, Lagrange’s equation of motion, in
which potential energy U depends on the generalized coordinate only, is described
as:


∂U
∂K
d ∂K
+
= u,
(A.6)
−
dt ∂ q˙g
∂qg
∂qg


hg2 (qg ,q̇g )Gg (qg )where the first term is calculated from (A.1):


d
d ∂K
= Jg (qg )q¨g + Jg (qg )q˙g .
dt ∂ q˙g
dt


(A.7)

hg1 (qg ,q̇g )Therefore, we obtain:
Jg (qg )q¨g + Hg (qg , q̇g ) + Gg (qg ) = u,

(A.8)

where Hg (qg , q̇g ) = (hg1 (qg , q̇g ) − hg2 (qg , q̇g )) is the non-linear term, which
includes the centrifugal force, and Coriolis force and Gg (qg ) is the gravity term.
They have the following decomposed structure:



0


Hg (qg , q̇g ) =  0  ∈ R 6 ,
H (ψ)

(A.9)

Back handspring of a multi-link gymnastic robot0Gg (qg ) =  Mg  ∈ R 6 .
0(4×1)

113

(A.10)

We do not show the each element of Hg (qg , q̇g ). It can be easily obtained by(A.5), (A.7) and (A.8), using any symbolic manipulation tool such as Maple V or
Mathematica. The joint-wise friction term W q̇g is not shown here, but is identified
and compensated for in the experiment.

ABOUT THE AUTHORS
Sang-Ho Hyon received the ME degree from Waseda Univerisity in 1998 and
the PhD degree from Tokyo Institute of Technology in 2002. He is currently an
Assistant Professor in the Department of Bioengineering and Robotics, Tohoku
University. His research interests are dynamic legged locomotion, non-linear
oscillation, etc. He is a member of the RSJ, SICE and IEEE.

Naoto Yokoyama received the ME degree from the Department of Bioengineering and Robotics, Tohoku University in 2004. He is currently with the Department
of EMD, Makino Corp. His research interest is robotic systems. He is a memberof the RSJ.

Takashi Emura received the PhD degree from Tohoku University in 1969. He iscurrently a Professor in the Department of Bioengineering and Robotics, Tohoku
University. During 1972–1973, he succeeded in walking a biped dynamically andfound that the semicircular canal should be defined as an angular velocity sensor.
His research interests include legged robots, automatic navigation vehicles, highprecision servomechanisms for numerically controlled machines, etc. He is amember of the RSJ, JSME, SICE, JSPE and IEEE.