Design and Evaluation of a Gravity Compensation ... - Semantic Scholar
Proceedings of the 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems San Diego, CA, USA, Oct 29 - Nov 2, 2007
ThC5.5
Design and Evaluation of a Gravity Compensation Mechanism for a Humanoid Robot Satoru Shirata, Atsushi Konno, and Masaru Uchiyama Knee torque
100
Single support phases 50 Torque [Nm]
Abstract— Performance of a human size humanoid robot is strictly limited by performance of the motor. The progress of a motor has not been remarkable compared with the progress of electronics. Therefore, the great progress of the performance of the motor cannot be expected, at least in the present circumstances. In this paper, a gravity compensation mechanism is designed which is applicable to a general biped robot. The mechanism is expected to reduce the joint torque of the legs required to support the gravitational force of the whole body. A humanoid robot Saika-4 is equipped with the gravity compensation mechanism in the legs. To evaluate effectiveness of the gravity compensation mechanism, preliminary experiments are performed using the humanoid robot.
Swing phases 0
-50
-100
Double support phases 0
Fig. 1.
1
2
3
4 Time [s]
5
6
7
8
Knee joint torque at a dynamic walking simulation of the HRP-2.
II. N ECCESITY OF G RAVITY C OMPENSATION I. I NTRODUCTION Recently, humanoid robots have been developed briskly and the technology of robots has been progressing remarkably. Nagasaka et al. realized a running of a humanoid robot SONY QRIO [1], which is the first running experiment of a humanoid robot in the world. Nagasaki et al. presented a running of a human size biped robot HRP-2LR [2]. HONDA Motor Co. Ltd. showed a running ability at 6 km/h of a humanoid robot ASIMO [3]. Nakaoka et al. demonstrated a Japanese traditional dance using a humanoid robot HRP-2 [4]. Uchiyama et al. demonstrated a drumbeating and a Japanese martial art Boujutsu using the HRP-2 in EXPO 2005, Aichi Japan [5]. However, the performance of the human size humanoid robots is strictly limited by the performance of the motor, because the progress of the motors has not been remarkable compared with the progress in the electronics. The ratio of motor power and motor weight is saturating, or at least the great progress cannot be expected. In order to overcome the limitation of performance, the load of motors must be reduced. A humanoid robot Saika-4 [6] is developed, in which a gravity compensation mechanism is employed using springs in the legs. This mechanism is expected to overcome the limitation of performance by reducing load of the motors. In this paper, the detail of the design of the gravity compensation mechanism is presented and the effectiveness of the mechanism is evaluated. Several motions of the humanoid robot are experimented to see the advantage of the gravity compensation mechanism. In these experiments, various conditions for the gravity compensation mechanism are examined. S. Shirata, A. Konno, and M. Uchiyama are with the Department of Aerospace Engineering, Tohoku University, 6-6-01 Aramakiaza-Aoba, Aoba-ku, Sendai 980-8579, Japan. {shirata, konno, uchiyama}@space.mech.tohoku.ac.jp
1-4244-0912-8/07/$25.00 ©2007 IEEE.
Among the loads for the joints, gravity is a dominant factor. The legs of a biped robot must support the gravitational force of the whole body in the support phase. On the other hand, the gravity affects on the legs so as to stretch the legs in the swing phase. However, since the mass of the legs are relatively smaller than the mass of the whole body, the torque needed for the swinging leg is theoretically considerably smaller than the torque needed for the supporting leg. Fig. 1 shows the joint torque of the right knee of a humanoid robot HRP-2 [7] obtained in a dynamic walking simulation using the OpenHRP [8]. As can be seen in Fig. 1, the knee joint torque in the swing phase is almost zero, while in the support phase the knee joint generates approximately −66 (Nm) to support the gravitational force of the whole body. Furthermore, in the static state before walking (t = 0.0 ∼ 1.7 (s)) and after walking (t = 6.3 ∼ 8.0 (s)), the offset torque of about −25 (Nm) is generated due to the gravitational force. If the gravitational load is compensated by a special mechanism, the joint torque will be balanced for positive and negative direction. As a result of the compensation, the power and the peak torque of the motors can be reduced, and hence, the performance of humanoid robots will be drastically improved. Especially when the humanoid robot works using the arms in the double support phase, the power consumption will be considerably decreased, since the stationary offset of torque is cancelled. III. G RAVITY C OMPENSATION M ECHANISM Some researchers have proposed gravity compensation mechanisms for robot arms (e.g. [9], [10]), however only few attempts have been made so far on compensation mechanisms for biped robots. Sugawara et al. proposed to use gas springs for torque reduction of the legs [11]. However, a parallel robot that does not have knees is used in [11], and hence, it is impossible to apply this mechanism to a general
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zB
zB xB
L2
yB
mg
mg
J1 J2 J3
L k4 a
L3 -θ 3 L4
J4 θ4
prA = [prAx prAy prAz ] (a) Side view
Pulley
θ2 J5 J6 T
θ4
W
fr
If the leg is designed so that L 3 = L4 = L, τ4s depends only on θ4 as follows:
Wire
fl
(b) Front view
Fig. 2. A 3-DOF kinematics model of legs.
compensates the offset torque generated in the static state such as the offset seen in Fig. 1 at t = 0.0 ∼ 1.7 (s) and t = 6.3 ∼ 8.0 (s). In such state, it is possible to approximate as prAx ≈ 0, prAy ≈ −W/2 and θ2 ≈ 0. Substituting the approximated states into Equations (3) and (4), the joint torque is given as follows: T mg T τ2s τ3s τ4s = 0 0 − L4 S34 (5) 2 θ4 mgL sin (6) 2 2 It is considered here to compensate τ 4s by a gravity compensation mechanism illustrated in Fig. 3. When the knee joint angle is θ 4 , the elongation of the spring becomes 2a sin(θ4 /2), where a is the length between the center of the knee joint and the pulley (see Fig. 3). It is assumed that the radius of the pulley is enough small compared with a. The compensation torque τ 4,comp is given as follows: τ4s = −
Fig. 3. Gravity compensation mechanism of knee pitch joint.
biped robot that has serial legs. Therefore, it is neccesary to design a gravity compensation mechanism applicable to general biped robots. A kinematic model of a biped robot is illustrated in Fig. 2. The axes of J1 ∼ J3 cross at a point, in the same way, axes of J5 and J6 cross at a point. J5 and J6 are assumed to be controlled so that a sole is parallel to the ground. J 1 is fixed on 0 (◦ ). It is assumed that the weight of the leg is smaller than the weight of the upper body enough to be ignored. The axes J5 and J6 cross at the point p rA with respect to the body coordinate system x B − yB − zB fixed on the center of gravity. p rA is given as follows: − (L3 S3 + L4 S34 ) prAx prAy = (L3 C3 + L4 C34 ) S2 − W , (1) 2 prAz − (L3 C3 + L4 C34 ) C2 − L2
τ4,comp = −2k4 a2 sin
0 J = 4(L3 C3 + L4 C34 ) C2 (L3 C3 + L4 C34 ) S2
− (L3 C3 + L4 C34 ) − (L3 S3 + L4 S34 ) S2 (L3 S3 + L4 S34 ) C2
(7)
where k4 is spring constant. The torque that the knee joint motor has to generate is given as follows: ∆τ4 = τ4s − τ4,comp θ4 mgL θ4 sin + 2k4 a2 sin =− 2 2 2 mgL θ4 sin = − (1 − µ4 ) 2 2 = (1 − µ4 ) τ4s , 4k4 a2 , µ4 = mgL
where Ci , Si , Cij and Sij represent cos θi , sin θi , cos (θi + θj ) and sin (θi + θj ), respectively. Differentiating Equation (1), the following equation is obtained. T T p˙ rAx p˙rAy p˙ rAz = J θ˙2 θ˙3 θ˙4 , (2) 2
θ4 , 2
(8) (9)
3 where µ4 stands for the ratio of the compensation. It should −L4 C34 be noticed that the ratio of the compensation µ 4 does not L4 S34 S2 5 . depend upon the knee joint angle θ . If the ratio of the 4 L4 S34 C2
fr and fl are ground reaction force generated at the center of the sole of the right leg and the left leg. From the balance of the forces and moments, a ground reaction force f r is calculated as follows: W + prAy mg, (3) fr = − W where the range of p rAy is −W ≦ prAy ≦ 0. When the ground reaction force generated, the joint torque is given as follows: τ2 0 (L3 C3 + L4 C34 ) S2 fr τ3 = J T 0 = (L3 S3 + L4 S34 ) C2 fr . (4) τ4 fr L4 S34 C2 fr Equation (4) is derived assuming that θ 1 is fixed, however, τ2 ∼ τ4 are functions of θ 1 ∼ θ4 in general. Therefore, it is almost impossible to design a mechanism that completely compensates for the gravitational force in all possible leg configuration. In this work, a mechanism is designed that
compensation µ 4 is set to 1, the gravity is completely compensated. The ratio µ 4 is adjustable by changing the parameter a or k 4 . IV. A NALYSIS OF THE GRAVITY COMPENSATION HRP-2
MECHANISM USING EXPERIMENTAL RESULT OF
A. Methods of analysis Experiments on the following three motions are performed at first using a humanoid robot HRP-2: (1) squatting down (bending knees) and standing up (stretching knees) on the ground, (2) bending and stretching knees in the air, and (3) dynamic walking. The motion (2) is the same motion as the motion (1), but in the air. Assuming that the humanoid robot HRP-2 is equipped with the gravity compensation mechanism discussed in Section III, joint torques during the three motions is calculated from the experimental data. Comparing the calculated data, the validity of the gravity compensation mechanism is evaluated. Furthermore, changing the ratio of gravity
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(a) Bending and stretching legs on the ground
120 100 80 60 θ4 40 Bending phase Stretching phase 20 0 -20 τ 4,comp -40 -60 0 2 4 6 8 10 12 14 Time [s]
100
Stretching phase
μ4=0.0 μ4=0.5 μ4=1.0
-50
μ4=0.0 μ4=0.5 μ4=1.0
100
τ 4,comp 0
1
2
3
4 5 Time [s]
6
7
8
(c) Dynamics walking
Single support phases
100
0
50
0
-50
-50
μ4=0.0 μ4=0.5 μ4=1.0
Single support phases
Swing phases
50 Torque [Nm]
0
Double support phases Swing phases
120 100 80 60 40 20 0 -20 -40 -60
Joint trajectories of the knee joint.
50 Torque [Nm]
Torque [Nm]
50
θ4
Torque [Nm]
μ4=0.0 μ4=0.5 μ4=1.0
Single support phases
(b) Bending and stretching legs in the air Fig. 4.
100
120 100 80 60 40 20 0 -20 -40 -60
Compensation torque [Nm]
120 100 80 60 40 20 0 -20 -40 -60
Compensation torque [Nm] Joint angle [°]
120 100 80 60 θ4 40 Bending phase Stretching phase 20 0 -20 τ 4,comp -40 -60 0 2 4 6 8 10 12 14 Time [s]
Compensation torque [Nm] Joint angle [°]
Joint angle [°]
120 100 80 60 40 20 0 -20 -40 -60
0
-50
Bending phase Bending phase
-100 0
2
4
6
8 Time [s]
Stretching phase
-100 10
12
14
0
(a) Bending/stretching on the ground Fig. 5.
2
4
6
8 Time [s]
10
-100 12
(b) Bending/stretching in the air
TABLE I Time (s) 0– 2
Phase 2 Phase 3
2– 6 6–10
Phase 4
10–14
4 Time [s]
5
6
7
8
3.0
(c) Dynamic walking
Swing phases 3.5
4.0 Time [s]
4.5
5.0
(d) Close up of (c)
From these experimental data, the joint torques are calculated assuming that the HRP-2 is equipped with the gravity compensation mechanism. The joint torque ∆τ 4,sim assuming the gravity compensation mechanism is given as follows: (10) ∆τ4,sim = τ4,exp − τ4,comp ,
Motion Keep initial position (the knee joint angle is 40 (◦ ) ) Squatting down 0.178 (m) with bending knees Keep squatting (the knee joint angle is 100 (◦ ) ) Standing up 0.178 (m) with stretching knees
where τ4,exp is the experimental data of the knee joint torque and τ4,comp is the compensation torque. From Equation (8), the compensation torque τ 4,comp is given as follows:
TABLE II Time (s) 0– 1 1– 7 7– 8
3
B. Computation of joint torque assuming the compensation
T HE DYNAMIC WALKING ( MOTION (1)). Phase 1 Phase 2 Phase 3
2
Calculated torque of the knee joint assuming the gravity compensation.
T HE BENDING AND STRETCHING KNEES ( MOTIONS (1) AND (2)). Phase 1
1
Double support phases
-100
Double support phases 0
14
Motion Keep initial position (θ4 = 40 (◦ )) Move forward 1 (m) by 4 step Keep standing
θ4 mgL sin , 2 2 where µ4 is the ratio of the compensation. τ4,comp = −µ4
compensation, the change of the joint torques are calculated to find an optimal ratio of compensation. The gravity compensation mechanism always generates the torque so as to stretch the legs. Therefore, the compensation torque affects as a load for the knee joints in the swing phase. The motion (2) (bending and stretching knees in the air) is performed to evaluate the adverse affect of the mechanism in the swing motion. From these analyses, the effectiveness of the gravity compensation mechanism in reducing gravitational torques is confirmed, and the optimum compensation ratio is selected. Table I shows the way of the bending and stretching knees, i.e. the motions (1) and (2). The HRP-2 starts to bend knees at t = 2 (s) and stops at t = 6 (s). During this period, the torso is moved 0.178 (m) lower than the initial position. Then the HRP-2 keeps the knee bending configuration until t = 10 (s). The HRP-2 starts to stretch the knees at t = 14 (s) to lift the torso up toward the initial position. Table II shows the way of the dynamic walking (motion (3)).
(11)
C. Experimental results and analyses Fig. 4 shows the joint trajectories of the knee joint and the compensation torque when the ratio of the compensation µ4 is 1.0. Fig. 5 (a)∼(d) show the calculated torque of the knee joint assuming the gravity compensation. In these analyses, three ratios of the compensation, µ 4 = 0.0, 0.5 and 1.0, are considered. The data with µ 4 = 0.0 are the actual experimental result, and others (µ 4 = 0.5 or 1.0) are calculated data assuming the gravity compensation. At first, the results of bending and stretching legs on the ground are analyzed (see Fig. 5 (a)). When there is no gravity compensation, the maximum torque is about 0 (Nm) and the minimum torque is about −70 (Nm). The motor of knee joint is always forced to produced a negative torque. On the other hand, when µ 4 is set to 1.0, the maximum torque is about 37 (Nm) and the minimum torque is about −32 (Nm). The motor torque is well balanced for positive
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TABLE III C OMPARISON OF TORQUES (N M ) ASSUMING THE GRAVITY COMPENSATION. Motion Stretching legs on the ground µ4 0.0 0.25 0.5 0.75 1.0 Average −34.5 −25.8 −17.1 −8.4 0.2 Absolute average 34.5 26.7 20.3 14.3 11.3 Maximum 0.0 8.8 18.2 27.5 36.8 −71.8 −61.3 −51.4 −41.4 −32.5 Minimum Absolute maximum 71.8 61.3 51.4 41.4 36.8 Max.+Min. −71.8 −52.5 −33.2 −13.9 4.4
Stretching legs in 0.0 0.25 0.5 4.8 13.4 22.1 15.0 17.9 23.2 35.9 44.2 53.0 −28.1 −20.7 −13.9 35.9 44.2 53.0 7.8 23.5 39.2
and negative directions. Furthermore, when the robot is in a stationary condition (at t = 0 ∼ 2, 6 ∼ 10 (s)), the gravitational torque is almost compensated. In the case of the bending and stretching legs in the air, the gravity compensation mechanism affects as a load for the knee joint as plotted in Fig. 5 (b). Next, the results of dynamic walking are considered (see Fig. 5 (d)). When there is no gravity compensation, the peak torque in the swing phase is about 60 (Nm) (at t =3.5 (s)) and the peak torque in the support phase is about −110 (Nm) (at t =4.8 (s)). The torque is biased to the negative direction. In contrast, when µ 4 is 1.0 the peak torque in the swing phase is about 90 (Nm) and the peak torque in the support phase is about −90 (Nm). The bias is cancelled and the torque is well balanced for positive and negative directions. In order to quantitatively evaluate the validity of the gravity compensation mechanism, the average, the absolute average, the maximum, the minimum, the absolute maximum and the sum of the maximum and the minimum of the torque in each motion is listed in Table III. These indices are computed for the cases in which µ 4 equals to 0.0, 0.25, 0.5, 0.75 and 1.0, in order to find an appropriate ratio. The meanings of each index of Table III are summerized here. The average shows the offset of the torque during the motion. The absolute average corresponds to the power consumption. The absolute maximum shows the peek value. This value is the most important index which determines the specification of the motor and the reduction gear. The sum of the maximum and the minimum shows the offset in the peek value. The values indicated by the bold italic typeface shows that the absolute of them is the smallest. In the case of bending/stretching on the ground and the dynamic walking, the absolute values of all indices become smallest when µ4 is 1.0. In the case of bending/stretching in the air, the absolute values of all indices become smallest when µ4 is 0.0 (without compensation). Fig. 1 and Fig. 5 (c) show the simulation result and the experimental result for the same dynamic walking. As can be seen in Fig. 1, torque in the swing phase is theoretically very small. However, as can be seen in Fig. 5 (c), the maximum torque of 61.6 (Nm) is generated in the swing phase, which cannot be ignored. When µ 4 is set to 1.0, the maximum torque becomes 89.3 (Nm). The possible reason of the torque difference in the swing phase between the simulation result and the experimental result would be un-modeled friction in the transmission. Considering of the unexpected effect in the swing phases, it would be better to set the ratio of the compensation to 0.5 rather than 1.0.
the air 0.75 1.0 30.8 39.4 30.9 39.4 61.8 70.7 −7.0 −0.8 61.8 70.7 54.8 69.9
Walking 0.0 0.25 0.5 0.75 1.0 −25.2 −18.8 −12.4 −6.0 0.4 36.7 32.9 29.2 25.6 22.6 61.6 68.6 75.5 82.4 89.3 −113.5 −107.6 −101.7 −95.9 −90.0 113.5 107.6 101.7 95.9 90.0 −51.8 −39.1 −26.3 −13.5 −0.7
Springs for gravity compensation Pulleys Wire
(a) CAD design Fig. 6.
(b) Inside of the thigh
Developed gravity compensation mechanism.
V. E XPERIMENTAL VERIFICATION OF THE GRAVITY COMPENSATION MECHANISM USING THE S AIKA -4 In this section, the experimental verification of the gravity compensation mechanism is described. From the consideration of the unexpected effect in the swing phases described in Section IV-C, the ratio of the compensation µ 4 is determined to 0.5. The gravity compensation mechanism of the Saika-4 is designed assuming that the mass of the upper body is 30 (kg). In order to put the gravity compensation mechanism into practical use, it is necessary to experimentally evaluate the effectiveness of the mechanism. Experiments are performed under various conditions using the humanoid robot Saika-4. A. Specifications of the Saika-4 The Saika-4 is developed as a self-sustaining, light-weight, and human-size humanoid robot. The height of the Saika-4 from sole to shoulder is 1.225 (m), and the total weight in the present state is 44 (kg). The weight of the leg is 13.7 (kg). The weights of the arm and hand are estimated to be 4 (kg) and 0.8 (kg). The total weight of the Saika-4 in the final stage is estimated to be 54 (kg). The Saika-4 has 6 degrees of freedom (DOF) at each leg, 7 DOF at each arm, 1 DOF at each hand and 2 DOF at robotics head, consequently, 30 DOF in total. Fig. 6 shows the designed and developed gravity compensation mechanism for the knee joints of the Saika-4. In the design of Saika-4 using 3D CAD modelling, the compensation mechanism was designed with the estimation of the weight of the upper body at 28 (kg). However, the
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TABLE IV
In the relatively slow motion, the joint velocity θ˙ is not very large, and thus, the joint torque can be approximated to be linear to the joint angle error:
D ETAIL OF THE STRETCHING MOTION . Phase Phase Phase Phase Phase
1 2 3 4 5
Time (s) 0–10 10–20 20–30 30–40 40–50
Motion Keep initial position (θ4 = 52 (◦ )) Lower the torso 0.3 (m) with bending knee Keep squatting (θ4 = 145 (◦ )) Raise the torso 0.3 (m) with stretching knee Keep standing
τ ≃ ΛK p θerr θerr = θref − θ
(15) (16)
arms and head are currently still under development and the actual weight is around 17 (kg), about 60% of the estimation of the total weight.
When the voltage is constant, the torque generating DC servo motor τ is given as follow:
B. Methods of experiments In the experiment, the effects of the gravity compensation mechanism are investigated in a motion of squatting down and standing up of the Saika-4. Table IV shows the experimental motion. The humanoid robot Saika-4 starts to bend the knees at t = 10 (s) and stops at t = 20 (s). During this period, the torso is moved 0.3 (m) lower than the initial position Then the Saika-4 keeps the knee bending configuration until t = 30 (s). The Saika-4 starts to stretch the knees at t = 30 (s) to lift the torso up toward the initial position. In order to clearly see the effectiveness of the gravity compensation removing the dynamic property, the Saika-4 moves slowly. In order to investigate how the gravity compensation mechanism affects the torque of the joints of the legs both in the support phase and swing phase, the squatting down and standing up experiments are performed on the ground and in the air. Therefore, the experiments are done under the following four conditions: 1) with the gravity compensation on the ground, 2) without the gravity compensation on the ground, 3) with the gravity compensation in the air, 4) without the gravity compensation in the air. The conditions 3) and 4) are assumed to simulate the motion in the swing phase. In the conditions 2) and 4), springs are removed from the mechanisms to see the difference in the results between with and without gravity compensation.
where K m is torque constant and I is current. Comparing Equation (15) with Equation (17), the current, I, can be approximate to be linear to the joint angle error:
C. Evaluation method For the motors attached to the joints of the Saika-4, velocity control type motor drivers are used. It is assumed that the joint torque is approximated by: (12) τ = Λ θ˙ ref − θ˙ ,
where Λ is a velocity feedback gain diagonal matrix, θ˙ com is the commanded joint reference velocity vector and θ˙ is the resultant joint velocity estimated by the governor circuits. In the experiments, the joint reference velocity is calculated by a simple proportional feedback control of the joint angles: (13) θ˙ com = K p (θref − θ) , where K p is a joint angle feedback gain diagonal matrix. Substituting Equation (13) into (12), the joint torque is expressed as: τ = Λ K p (θref − θ) − θ˙ , (14)
τ = K m I,
I≃
ΛK p θerr Km
(17)
(18)
Total power output of the motor P is given as follow:
tend P = Ra I 2 dt, (19) t0
where Ra is Terminal resistance of the motor . Substituting Equation (18) into (19), the total power of the motor can be approximate to be linear to the square of the joint angle error: 2 tend ΛK p {θerr }2 dt (20) P ≃ Ra Km t0 Using Current monitoring function of motor driver, it is possible to estimate the joint torque of the Saika-4. However, since there are many noises, it is difficult to measure the joint torque precisely at this experiment. Therefore, the joint angle error is used to evaluate the effect of the gravity compensation. D. Experimental Result and Discussion Fig. 7 shows the joint angle error θ err (◦ ) in the experiment. Fig. 7 (a) and (b) show the result for the condition 1) and 2) and the result for the condition 3) and 4), respectively. Table V shows the average, the absolute average, the maximum, the minimum, the absolute maximum and the sum of the maximum and the minimum of the joint angle error. In addition, these show the power normalized by the result of the condition without gravity compensation on the ground. As shown in Fig. 7 (a), remarkable improvement in the gravitational load can be seen in the knee joint when the gravity compensation mechanism is used. The result of no gravity compensation has a positive offset, while the result of gravity compensation shows the equality for the positive and negative direction. In addition, the absolute value of the knee joint angle error is reduced when the gravity compensation is applied. This reduction is the main advantage of the gravity compensation. This advantage in the knee joint is also seen in the Table V. The performance of the motor is limited by the absolute value of the maximum required torque. The gravity compensation mechanism shifts the torque required for the knee joints to the negative direction and contributes to equalize the
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0.2
0.2
With gravity compensation Without gravity compensation
0.15
0.1
Angle deflection [°]
Angle deflection [°]
0.15
0.05 0 -0.05
0 -0.05
-0.1
-0.1 -0.15
0
10
20
30
40
50
-0.2
0
10
Time [s]
20
30
40
50
Time [s]
(a) Stretching legs on the ground Fig. 7.
This paper described the design of the gravity compensation mechanisum, analyses of the validity using the experimental result of the HRP-2, and the experimental evaluation for the gravity compensation mechanism. From the analyses using the experimental result of the humanoid robot HRP-2, the effectiveness of the gravity compensation mechanism in reducing gravitational torques was confirmed, and the compensation ratio was specified to be 0.5 in the double support phase. In order to verify the effects of the gravity compensation mechanism, the squatting down and standing up motions of the humanoid robot Saika-4 were experimented. To evaluate the performance of leg motion with and without the gravity compensation mechanism, and to evaluate the effect of the mechanism in the support phase and the swing phase, it was experimented under the four conditions. The joint angle errors were compared in each condition.
0.1 0.05
-0.15 -0.2
VI. CONCLUSION
With gravity compensation Without gravity compensation
(b) Stretching legs in the air
Knee joint angle error with and without gravity compensation.
TABLE V K NEE JOINT ANGLE ERROR ([◦ ]) AT STRETCHING MOTION . Experiment No. Average Absolute average Maximum Minimum Absolute maximum Max. + Min. Nolmarized power
1) 0.0135 0.0365 0.1248 -0.0957 0.1248 0.0291 0.4301
2) 0.0599 0.0650 0.1738 -0.0410 0.1738 0.1328 1.0000
3) -0.0508 0.0585 0.0599 -0.1812 0.1812 -0.1213 0.9045
4) -0.0211 0.0421 0.0941 -0.1515 0.1515 -0.0574 0.6235
ACKNOWLEDGMENT
torque for positive and negative direction. Consequently, the absolute value of the maximum required torque is reduced when the gravity compensation is applied. Next, the results when the robot moves in the air are evaluated. This movement is assumed to simulate the swing phase of the legs. Fig. 7 (b) shows that angle errors become slightly larger when a gravity compensation is applied. This result indicates that the gravity compensation may act as a disturbance for the joint control in the swing phase. Using Table V, the effectiveness of the gravity compensation mechanism is evaluated quantitatively. Comparing condition 1) with condition 2), the absolute average, the absolute max and the total power are reduced to 56%, 72% and 43% respectively by the gravity compensation. The absolute average and the total power for condition 3) is smaller about 10% than these indexes at condition 2). This result shows that the negative effects of the mechanism in the swing phase is not large. As described in the beginning of this chapter, the head and arms are still under development. Therefore, the total weight of the upper body is currently only 60% of the expected weight during the designing stage. As a result, in the present state the compensation ratio is at 0.83 ≈ 0.5/0.6, larger than the expected ratio of 0.5. Therefore, as is shown in Fig. 7 (a), the compensated torque during steady state rises to 83%, causing the joint angle error to be close to 0 ( ◦ ). The gravity compensation mechanism just gives a bias torque to joints, and thus, the effect is not affected by a speed of movement in theory. However, the influence of the motion speed to the gravity compensation must be carefully investigated in future. In the experiments, the joint angle errors are used to evaluate the effect of the gravity compensation mechanism. This method is insufficient for a quantitative evaluation. Therefore, the electric current of the motor will be used for the evaluation in the future work.
This research was supported by NEDO Industrial Technology Research Grant Program (project ID: 05A30703a). R EFERENCES [1] K. Nagasaka, Y. Kuroki, S. Suzuki, Y. Itoh, and J. Yamaguchi, “Integrated motion control for walking, jumping and running on a small biped entertainment robot,” in Proc. of IEEE Int. Conf. on Robotics and Automation, New Orleans, USA, April 2004, pp. 3189– 3194. [2] T. Nagasaki, S. Kajita, K. Kaneko, K. Yokoi, and K. Tanie, “A runnig experiment of humanoid biped,” in Proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Sendai, Japan, September 2004, pp. 136–141. [3] “http://world.honda.com/asimo/.” [4] S. Nakaoka, A. Nakazawa, F. Kanahiro, K. Kaneko, M. Morisawa, and K. Ikeuchi, “Task model of lower body motion for a biped humanoid robot to imitate human dances,” in Proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Edmonton, Alberta, Canada, August 2005, pp. 2769–2774. [5] M. Uchiyama, A. Konno, D. Sato, T. Matsumoto, Y. Ishida, and T. Ishihara, “A humanoid robot to perform tasks in which impact dynamics is utilized,” in Proc. of 36th Int. Symp. on Robotics, Tokyo, Japan, November 2005, p. TH1C1. [6] S. Shirata, A. Konno, and M. Uchiyama, “Design and development of a light-weight biped humanoid robot saika-4,” in Proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Sendai, Japan, September 2004, pp. 148–153. [7] K. Kaneko, F. Kanehiro, S. kajita, H. Hirukawa, T. Kawasaki, M. Hurata, K. Akachi, and T. Isozumi, “Humanoid robot hrp-2,” in Proc. of IEEE Int. Conf. on Robotics and Automation, New Orleans, USA, April 2004, pp. 1083–1090. [8] H. Hirukawa, F. Kanehiro, and S. Kajita, Open HRP: Open Architecture Humanoid Robotics Platform, R. A. Jarvis and A. Zelinsky, Eds. Springer-Verlag, 2003. [9] S. Hirose and R. Chu, “Development of a lightweight torque limiting m-drive actuator for hyper-redundant manipulator float arm,” in Proc. of IEEE Int. Conf. on Robotics and Automation, Detroit, USA, May 1999, pp. 2831–2836. [10] T. Morita, F. Kuribara, Y. Shiozawa, and S. Sugano, “A novel mechanism design for gravity compensation in three dimensional space,” in Proc. of IEEE/ASME Int. Conf. on Advanced Intelligent Mechatronics, Paper No. 280, 2003. [11] Y. Sugahara, Y. Mikuriya, T. Hosobata, H. Sunazuka, M. Kawase, K. Hashimoto, H. ok Lim, and A. Takanishi, “Support torque reduction mechanism for biped locomotor with parallel mechanism,” in Proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Sendai, Japan, September 2004, pp. 3213–3218.
3640
ThC5.5
Design and Evaluation of a Gravity Compensation Mechanism for a Humanoid Robot Satoru Shirata, Atsushi Konno, and Masaru Uchiyama Knee torque
100
Single support phases 50 Torque [Nm]
Abstract— Performance of a human size humanoid robot is strictly limited by performance of the motor. The progress of a motor has not been remarkable compared with the progress of electronics. Therefore, the great progress of the performance of the motor cannot be expected, at least in the present circumstances. In this paper, a gravity compensation mechanism is designed which is applicable to a general biped robot. The mechanism is expected to reduce the joint torque of the legs required to support the gravitational force of the whole body. A humanoid robot Saika-4 is equipped with the gravity compensation mechanism in the legs. To evaluate effectiveness of the gravity compensation mechanism, preliminary experiments are performed using the humanoid robot.
Swing phases 0
-50
-100
Double support phases 0
Fig. 1.
1
2
3
4 Time [s]
5
6
7
8
Knee joint torque at a dynamic walking simulation of the HRP-2.
II. N ECCESITY OF G RAVITY C OMPENSATION I. I NTRODUCTION Recently, humanoid robots have been developed briskly and the technology of robots has been progressing remarkably. Nagasaka et al. realized a running of a humanoid robot SONY QRIO [1], which is the first running experiment of a humanoid robot in the world. Nagasaki et al. presented a running of a human size biped robot HRP-2LR [2]. HONDA Motor Co. Ltd. showed a running ability at 6 km/h of a humanoid robot ASIMO [3]. Nakaoka et al. demonstrated a Japanese traditional dance using a humanoid robot HRP-2 [4]. Uchiyama et al. demonstrated a drumbeating and a Japanese martial art Boujutsu using the HRP-2 in EXPO 2005, Aichi Japan [5]. However, the performance of the human size humanoid robots is strictly limited by the performance of the motor, because the progress of the motors has not been remarkable compared with the progress in the electronics. The ratio of motor power and motor weight is saturating, or at least the great progress cannot be expected. In order to overcome the limitation of performance, the load of motors must be reduced. A humanoid robot Saika-4 [6] is developed, in which a gravity compensation mechanism is employed using springs in the legs. This mechanism is expected to overcome the limitation of performance by reducing load of the motors. In this paper, the detail of the design of the gravity compensation mechanism is presented and the effectiveness of the mechanism is evaluated. Several motions of the humanoid robot are experimented to see the advantage of the gravity compensation mechanism. In these experiments, various conditions for the gravity compensation mechanism are examined. S. Shirata, A. Konno, and M. Uchiyama are with the Department of Aerospace Engineering, Tohoku University, 6-6-01 Aramakiaza-Aoba, Aoba-ku, Sendai 980-8579, Japan. {shirata, konno, uchiyama}@space.mech.tohoku.ac.jp
1-4244-0912-8/07/$25.00 ©2007 IEEE.
Among the loads for the joints, gravity is a dominant factor. The legs of a biped robot must support the gravitational force of the whole body in the support phase. On the other hand, the gravity affects on the legs so as to stretch the legs in the swing phase. However, since the mass of the legs are relatively smaller than the mass of the whole body, the torque needed for the swinging leg is theoretically considerably smaller than the torque needed for the supporting leg. Fig. 1 shows the joint torque of the right knee of a humanoid robot HRP-2 [7] obtained in a dynamic walking simulation using the OpenHRP [8]. As can be seen in Fig. 1, the knee joint torque in the swing phase is almost zero, while in the support phase the knee joint generates approximately −66 (Nm) to support the gravitational force of the whole body. Furthermore, in the static state before walking (t = 0.0 ∼ 1.7 (s)) and after walking (t = 6.3 ∼ 8.0 (s)), the offset torque of about −25 (Nm) is generated due to the gravitational force. If the gravitational load is compensated by a special mechanism, the joint torque will be balanced for positive and negative direction. As a result of the compensation, the power and the peak torque of the motors can be reduced, and hence, the performance of humanoid robots will be drastically improved. Especially when the humanoid robot works using the arms in the double support phase, the power consumption will be considerably decreased, since the stationary offset of torque is cancelled. III. G RAVITY C OMPENSATION M ECHANISM Some researchers have proposed gravity compensation mechanisms for robot arms (e.g. [9], [10]), however only few attempts have been made so far on compensation mechanisms for biped robots. Sugawara et al. proposed to use gas springs for torque reduction of the legs [11]. However, a parallel robot that does not have knees is used in [11], and hence, it is impossible to apply this mechanism to a general
3635
zB
zB xB
L2
yB
mg
mg
J1 J2 J3
L k4 a
L3 -θ 3 L4
J4 θ4
prA = [prAx prAy prAz ] (a) Side view
Pulley
θ2 J5 J6 T
θ4
W
fr
If the leg is designed so that L 3 = L4 = L, τ4s depends only on θ4 as follows:
Wire
fl
(b) Front view
Fig. 2. A 3-DOF kinematics model of legs.
compensates the offset torque generated in the static state such as the offset seen in Fig. 1 at t = 0.0 ∼ 1.7 (s) and t = 6.3 ∼ 8.0 (s). In such state, it is possible to approximate as prAx ≈ 0, prAy ≈ −W/2 and θ2 ≈ 0. Substituting the approximated states into Equations (3) and (4), the joint torque is given as follows: T mg T τ2s τ3s τ4s = 0 0 − L4 S34 (5) 2 θ4 mgL sin (6) 2 2 It is considered here to compensate τ 4s by a gravity compensation mechanism illustrated in Fig. 3. When the knee joint angle is θ 4 , the elongation of the spring becomes 2a sin(θ4 /2), where a is the length between the center of the knee joint and the pulley (see Fig. 3). It is assumed that the radius of the pulley is enough small compared with a. The compensation torque τ 4,comp is given as follows: τ4s = −
Fig. 3. Gravity compensation mechanism of knee pitch joint.
biped robot that has serial legs. Therefore, it is neccesary to design a gravity compensation mechanism applicable to general biped robots. A kinematic model of a biped robot is illustrated in Fig. 2. The axes of J1 ∼ J3 cross at a point, in the same way, axes of J5 and J6 cross at a point. J5 and J6 are assumed to be controlled so that a sole is parallel to the ground. J 1 is fixed on 0 (◦ ). It is assumed that the weight of the leg is smaller than the weight of the upper body enough to be ignored. The axes J5 and J6 cross at the point p rA with respect to the body coordinate system x B − yB − zB fixed on the center of gravity. p rA is given as follows: − (L3 S3 + L4 S34 ) prAx prAy = (L3 C3 + L4 C34 ) S2 − W , (1) 2 prAz − (L3 C3 + L4 C34 ) C2 − L2
τ4,comp = −2k4 a2 sin
0 J = 4(L3 C3 + L4 C34 ) C2 (L3 C3 + L4 C34 ) S2
− (L3 C3 + L4 C34 ) − (L3 S3 + L4 S34 ) S2 (L3 S3 + L4 S34 ) C2
(7)
where k4 is spring constant. The torque that the knee joint motor has to generate is given as follows: ∆τ4 = τ4s − τ4,comp θ4 mgL θ4 sin + 2k4 a2 sin =− 2 2 2 mgL θ4 sin = − (1 − µ4 ) 2 2 = (1 − µ4 ) τ4s , 4k4 a2 , µ4 = mgL
where Ci , Si , Cij and Sij represent cos θi , sin θi , cos (θi + θj ) and sin (θi + θj ), respectively. Differentiating Equation (1), the following equation is obtained. T T p˙ rAx p˙rAy p˙ rAz = J θ˙2 θ˙3 θ˙4 , (2) 2
θ4 , 2
(8) (9)
3 where µ4 stands for the ratio of the compensation. It should −L4 C34 be noticed that the ratio of the compensation µ 4 does not L4 S34 S2 5 . depend upon the knee joint angle θ . If the ratio of the 4 L4 S34 C2
fr and fl are ground reaction force generated at the center of the sole of the right leg and the left leg. From the balance of the forces and moments, a ground reaction force f r is calculated as follows: W + prAy mg, (3) fr = − W where the range of p rAy is −W ≦ prAy ≦ 0. When the ground reaction force generated, the joint torque is given as follows: τ2 0 (L3 C3 + L4 C34 ) S2 fr τ3 = J T 0 = (L3 S3 + L4 S34 ) C2 fr . (4) τ4 fr L4 S34 C2 fr Equation (4) is derived assuming that θ 1 is fixed, however, τ2 ∼ τ4 are functions of θ 1 ∼ θ4 in general. Therefore, it is almost impossible to design a mechanism that completely compensates for the gravitational force in all possible leg configuration. In this work, a mechanism is designed that
compensation µ 4 is set to 1, the gravity is completely compensated. The ratio µ 4 is adjustable by changing the parameter a or k 4 . IV. A NALYSIS OF THE GRAVITY COMPENSATION HRP-2
MECHANISM USING EXPERIMENTAL RESULT OF
A. Methods of analysis Experiments on the following three motions are performed at first using a humanoid robot HRP-2: (1) squatting down (bending knees) and standing up (stretching knees) on the ground, (2) bending and stretching knees in the air, and (3) dynamic walking. The motion (2) is the same motion as the motion (1), but in the air. Assuming that the humanoid robot HRP-2 is equipped with the gravity compensation mechanism discussed in Section III, joint torques during the three motions is calculated from the experimental data. Comparing the calculated data, the validity of the gravity compensation mechanism is evaluated. Furthermore, changing the ratio of gravity
3636
(a) Bending and stretching legs on the ground
120 100 80 60 θ4 40 Bending phase Stretching phase 20 0 -20 τ 4,comp -40 -60 0 2 4 6 8 10 12 14 Time [s]
100
Stretching phase
μ4=0.0 μ4=0.5 μ4=1.0
-50
μ4=0.0 μ4=0.5 μ4=1.0
100
τ 4,comp 0
1
2
3
4 5 Time [s]
6
7
8
(c) Dynamics walking
Single support phases
100
0
50
0
-50
-50
μ4=0.0 μ4=0.5 μ4=1.0
Single support phases
Swing phases
50 Torque [Nm]
0
Double support phases Swing phases
120 100 80 60 40 20 0 -20 -40 -60
Joint trajectories of the knee joint.
50 Torque [Nm]
Torque [Nm]
50
θ4
Torque [Nm]
μ4=0.0 μ4=0.5 μ4=1.0
Single support phases
(b) Bending and stretching legs in the air Fig. 4.
100
120 100 80 60 40 20 0 -20 -40 -60
Compensation torque [Nm]
120 100 80 60 40 20 0 -20 -40 -60
Compensation torque [Nm] Joint angle [°]
120 100 80 60 θ4 40 Bending phase Stretching phase 20 0 -20 τ 4,comp -40 -60 0 2 4 6 8 10 12 14 Time [s]
Compensation torque [Nm] Joint angle [°]
Joint angle [°]
120 100 80 60 40 20 0 -20 -40 -60
0
-50
Bending phase Bending phase
-100 0
2
4
6
8 Time [s]
Stretching phase
-100 10
12
14
0
(a) Bending/stretching on the ground Fig. 5.
2
4
6
8 Time [s]
10
-100 12
(b) Bending/stretching in the air
TABLE I Time (s) 0– 2
Phase 2 Phase 3
2– 6 6–10
Phase 4
10–14
4 Time [s]
5
6
7
8
3.0
(c) Dynamic walking
Swing phases 3.5
4.0 Time [s]
4.5
5.0
(d) Close up of (c)
From these experimental data, the joint torques are calculated assuming that the HRP-2 is equipped with the gravity compensation mechanism. The joint torque ∆τ 4,sim assuming the gravity compensation mechanism is given as follows: (10) ∆τ4,sim = τ4,exp − τ4,comp ,
Motion Keep initial position (the knee joint angle is 40 (◦ ) ) Squatting down 0.178 (m) with bending knees Keep squatting (the knee joint angle is 100 (◦ ) ) Standing up 0.178 (m) with stretching knees
where τ4,exp is the experimental data of the knee joint torque and τ4,comp is the compensation torque. From Equation (8), the compensation torque τ 4,comp is given as follows:
TABLE II Time (s) 0– 1 1– 7 7– 8
3
B. Computation of joint torque assuming the compensation
T HE DYNAMIC WALKING ( MOTION (1)). Phase 1 Phase 2 Phase 3
2
Calculated torque of the knee joint assuming the gravity compensation.
T HE BENDING AND STRETCHING KNEES ( MOTIONS (1) AND (2)). Phase 1
1
Double support phases
-100
Double support phases 0
14
Motion Keep initial position (θ4 = 40 (◦ )) Move forward 1 (m) by 4 step Keep standing
θ4 mgL sin , 2 2 where µ4 is the ratio of the compensation. τ4,comp = −µ4
compensation, the change of the joint torques are calculated to find an optimal ratio of compensation. The gravity compensation mechanism always generates the torque so as to stretch the legs. Therefore, the compensation torque affects as a load for the knee joints in the swing phase. The motion (2) (bending and stretching knees in the air) is performed to evaluate the adverse affect of the mechanism in the swing motion. From these analyses, the effectiveness of the gravity compensation mechanism in reducing gravitational torques is confirmed, and the optimum compensation ratio is selected. Table I shows the way of the bending and stretching knees, i.e. the motions (1) and (2). The HRP-2 starts to bend knees at t = 2 (s) and stops at t = 6 (s). During this period, the torso is moved 0.178 (m) lower than the initial position. Then the HRP-2 keeps the knee bending configuration until t = 10 (s). The HRP-2 starts to stretch the knees at t = 14 (s) to lift the torso up toward the initial position. Table II shows the way of the dynamic walking (motion (3)).
(11)
C. Experimental results and analyses Fig. 4 shows the joint trajectories of the knee joint and the compensation torque when the ratio of the compensation µ4 is 1.0. Fig. 5 (a)∼(d) show the calculated torque of the knee joint assuming the gravity compensation. In these analyses, three ratios of the compensation, µ 4 = 0.0, 0.5 and 1.0, are considered. The data with µ 4 = 0.0 are the actual experimental result, and others (µ 4 = 0.5 or 1.0) are calculated data assuming the gravity compensation. At first, the results of bending and stretching legs on the ground are analyzed (see Fig. 5 (a)). When there is no gravity compensation, the maximum torque is about 0 (Nm) and the minimum torque is about −70 (Nm). The motor of knee joint is always forced to produced a negative torque. On the other hand, when µ 4 is set to 1.0, the maximum torque is about 37 (Nm) and the minimum torque is about −32 (Nm). The motor torque is well balanced for positive
3637
TABLE III C OMPARISON OF TORQUES (N M ) ASSUMING THE GRAVITY COMPENSATION. Motion Stretching legs on the ground µ4 0.0 0.25 0.5 0.75 1.0 Average −34.5 −25.8 −17.1 −8.4 0.2 Absolute average 34.5 26.7 20.3 14.3 11.3 Maximum 0.0 8.8 18.2 27.5 36.8 −71.8 −61.3 −51.4 −41.4 −32.5 Minimum Absolute maximum 71.8 61.3 51.4 41.4 36.8 Max.+Min. −71.8 −52.5 −33.2 −13.9 4.4
Stretching legs in 0.0 0.25 0.5 4.8 13.4 22.1 15.0 17.9 23.2 35.9 44.2 53.0 −28.1 −20.7 −13.9 35.9 44.2 53.0 7.8 23.5 39.2
and negative directions. Furthermore, when the robot is in a stationary condition (at t = 0 ∼ 2, 6 ∼ 10 (s)), the gravitational torque is almost compensated. In the case of the bending and stretching legs in the air, the gravity compensation mechanism affects as a load for the knee joint as plotted in Fig. 5 (b). Next, the results of dynamic walking are considered (see Fig. 5 (d)). When there is no gravity compensation, the peak torque in the swing phase is about 60 (Nm) (at t =3.5 (s)) and the peak torque in the support phase is about −110 (Nm) (at t =4.8 (s)). The torque is biased to the negative direction. In contrast, when µ 4 is 1.0 the peak torque in the swing phase is about 90 (Nm) and the peak torque in the support phase is about −90 (Nm). The bias is cancelled and the torque is well balanced for positive and negative directions. In order to quantitatively evaluate the validity of the gravity compensation mechanism, the average, the absolute average, the maximum, the minimum, the absolute maximum and the sum of the maximum and the minimum of the torque in each motion is listed in Table III. These indices are computed for the cases in which µ 4 equals to 0.0, 0.25, 0.5, 0.75 and 1.0, in order to find an appropriate ratio. The meanings of each index of Table III are summerized here. The average shows the offset of the torque during the motion. The absolute average corresponds to the power consumption. The absolute maximum shows the peek value. This value is the most important index which determines the specification of the motor and the reduction gear. The sum of the maximum and the minimum shows the offset in the peek value. The values indicated by the bold italic typeface shows that the absolute of them is the smallest. In the case of bending/stretching on the ground and the dynamic walking, the absolute values of all indices become smallest when µ4 is 1.0. In the case of bending/stretching in the air, the absolute values of all indices become smallest when µ4 is 0.0 (without compensation). Fig. 1 and Fig. 5 (c) show the simulation result and the experimental result for the same dynamic walking. As can be seen in Fig. 1, torque in the swing phase is theoretically very small. However, as can be seen in Fig. 5 (c), the maximum torque of 61.6 (Nm) is generated in the swing phase, which cannot be ignored. When µ 4 is set to 1.0, the maximum torque becomes 89.3 (Nm). The possible reason of the torque difference in the swing phase between the simulation result and the experimental result would be un-modeled friction in the transmission. Considering of the unexpected effect in the swing phases, it would be better to set the ratio of the compensation to 0.5 rather than 1.0.
the air 0.75 1.0 30.8 39.4 30.9 39.4 61.8 70.7 −7.0 −0.8 61.8 70.7 54.8 69.9
Walking 0.0 0.25 0.5 0.75 1.0 −25.2 −18.8 −12.4 −6.0 0.4 36.7 32.9 29.2 25.6 22.6 61.6 68.6 75.5 82.4 89.3 −113.5 −107.6 −101.7 −95.9 −90.0 113.5 107.6 101.7 95.9 90.0 −51.8 −39.1 −26.3 −13.5 −0.7
Springs for gravity compensation Pulleys Wire
(a) CAD design Fig. 6.
(b) Inside of the thigh
Developed gravity compensation mechanism.
V. E XPERIMENTAL VERIFICATION OF THE GRAVITY COMPENSATION MECHANISM USING THE S AIKA -4 In this section, the experimental verification of the gravity compensation mechanism is described. From the consideration of the unexpected effect in the swing phases described in Section IV-C, the ratio of the compensation µ 4 is determined to 0.5. The gravity compensation mechanism of the Saika-4 is designed assuming that the mass of the upper body is 30 (kg). In order to put the gravity compensation mechanism into practical use, it is necessary to experimentally evaluate the effectiveness of the mechanism. Experiments are performed under various conditions using the humanoid robot Saika-4. A. Specifications of the Saika-4 The Saika-4 is developed as a self-sustaining, light-weight, and human-size humanoid robot. The height of the Saika-4 from sole to shoulder is 1.225 (m), and the total weight in the present state is 44 (kg). The weight of the leg is 13.7 (kg). The weights of the arm and hand are estimated to be 4 (kg) and 0.8 (kg). The total weight of the Saika-4 in the final stage is estimated to be 54 (kg). The Saika-4 has 6 degrees of freedom (DOF) at each leg, 7 DOF at each arm, 1 DOF at each hand and 2 DOF at robotics head, consequently, 30 DOF in total. Fig. 6 shows the designed and developed gravity compensation mechanism for the knee joints of the Saika-4. In the design of Saika-4 using 3D CAD modelling, the compensation mechanism was designed with the estimation of the weight of the upper body at 28 (kg). However, the
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TABLE IV
In the relatively slow motion, the joint velocity θ˙ is not very large, and thus, the joint torque can be approximated to be linear to the joint angle error:
D ETAIL OF THE STRETCHING MOTION . Phase Phase Phase Phase Phase
1 2 3 4 5
Time (s) 0–10 10–20 20–30 30–40 40–50
Motion Keep initial position (θ4 = 52 (◦ )) Lower the torso 0.3 (m) with bending knee Keep squatting (θ4 = 145 (◦ )) Raise the torso 0.3 (m) with stretching knee Keep standing
τ ≃ ΛK p θerr θerr = θref − θ
(15) (16)
arms and head are currently still under development and the actual weight is around 17 (kg), about 60% of the estimation of the total weight.
When the voltage is constant, the torque generating DC servo motor τ is given as follow:
B. Methods of experiments In the experiment, the effects of the gravity compensation mechanism are investigated in a motion of squatting down and standing up of the Saika-4. Table IV shows the experimental motion. The humanoid robot Saika-4 starts to bend the knees at t = 10 (s) and stops at t = 20 (s). During this period, the torso is moved 0.3 (m) lower than the initial position Then the Saika-4 keeps the knee bending configuration until t = 30 (s). The Saika-4 starts to stretch the knees at t = 30 (s) to lift the torso up toward the initial position. In order to clearly see the effectiveness of the gravity compensation removing the dynamic property, the Saika-4 moves slowly. In order to investigate how the gravity compensation mechanism affects the torque of the joints of the legs both in the support phase and swing phase, the squatting down and standing up experiments are performed on the ground and in the air. Therefore, the experiments are done under the following four conditions: 1) with the gravity compensation on the ground, 2) without the gravity compensation on the ground, 3) with the gravity compensation in the air, 4) without the gravity compensation in the air. The conditions 3) and 4) are assumed to simulate the motion in the swing phase. In the conditions 2) and 4), springs are removed from the mechanisms to see the difference in the results between with and without gravity compensation.
where K m is torque constant and I is current. Comparing Equation (15) with Equation (17), the current, I, can be approximate to be linear to the joint angle error:
C. Evaluation method For the motors attached to the joints of the Saika-4, velocity control type motor drivers are used. It is assumed that the joint torque is approximated by: (12) τ = Λ θ˙ ref − θ˙ ,
where Λ is a velocity feedback gain diagonal matrix, θ˙ com is the commanded joint reference velocity vector and θ˙ is the resultant joint velocity estimated by the governor circuits. In the experiments, the joint reference velocity is calculated by a simple proportional feedback control of the joint angles: (13) θ˙ com = K p (θref − θ) , where K p is a joint angle feedback gain diagonal matrix. Substituting Equation (13) into (12), the joint torque is expressed as: τ = Λ K p (θref − θ) − θ˙ , (14)
τ = K m I,
I≃
ΛK p θerr Km
(17)
(18)
Total power output of the motor P is given as follow:
tend P = Ra I 2 dt, (19) t0
where Ra is Terminal resistance of the motor . Substituting Equation (18) into (19), the total power of the motor can be approximate to be linear to the square of the joint angle error: 2 tend ΛK p {θerr }2 dt (20) P ≃ Ra Km t0 Using Current monitoring function of motor driver, it is possible to estimate the joint torque of the Saika-4. However, since there are many noises, it is difficult to measure the joint torque precisely at this experiment. Therefore, the joint angle error is used to evaluate the effect of the gravity compensation. D. Experimental Result and Discussion Fig. 7 shows the joint angle error θ err (◦ ) in the experiment. Fig. 7 (a) and (b) show the result for the condition 1) and 2) and the result for the condition 3) and 4), respectively. Table V shows the average, the absolute average, the maximum, the minimum, the absolute maximum and the sum of the maximum and the minimum of the joint angle error. In addition, these show the power normalized by the result of the condition without gravity compensation on the ground. As shown in Fig. 7 (a), remarkable improvement in the gravitational load can be seen in the knee joint when the gravity compensation mechanism is used. The result of no gravity compensation has a positive offset, while the result of gravity compensation shows the equality for the positive and negative direction. In addition, the absolute value of the knee joint angle error is reduced when the gravity compensation is applied. This reduction is the main advantage of the gravity compensation. This advantage in the knee joint is also seen in the Table V. The performance of the motor is limited by the absolute value of the maximum required torque. The gravity compensation mechanism shifts the torque required for the knee joints to the negative direction and contributes to equalize the
3639
0.2
0.2
With gravity compensation Without gravity compensation
0.15
0.1
Angle deflection [°]
Angle deflection [°]
0.15
0.05 0 -0.05
0 -0.05
-0.1
-0.1 -0.15
0
10
20
30
40
50
-0.2
0
10
Time [s]
20
30
40
50
Time [s]
(a) Stretching legs on the ground Fig. 7.
This paper described the design of the gravity compensation mechanisum, analyses of the validity using the experimental result of the HRP-2, and the experimental evaluation for the gravity compensation mechanism. From the analyses using the experimental result of the humanoid robot HRP-2, the effectiveness of the gravity compensation mechanism in reducing gravitational torques was confirmed, and the compensation ratio was specified to be 0.5 in the double support phase. In order to verify the effects of the gravity compensation mechanism, the squatting down and standing up motions of the humanoid robot Saika-4 were experimented. To evaluate the performance of leg motion with and without the gravity compensation mechanism, and to evaluate the effect of the mechanism in the support phase and the swing phase, it was experimented under the four conditions. The joint angle errors were compared in each condition.
0.1 0.05
-0.15 -0.2
VI. CONCLUSION
With gravity compensation Without gravity compensation
(b) Stretching legs in the air
Knee joint angle error with and without gravity compensation.
TABLE V K NEE JOINT ANGLE ERROR ([◦ ]) AT STRETCHING MOTION . Experiment No. Average Absolute average Maximum Minimum Absolute maximum Max. + Min. Nolmarized power
1) 0.0135 0.0365 0.1248 -0.0957 0.1248 0.0291 0.4301
2) 0.0599 0.0650 0.1738 -0.0410 0.1738 0.1328 1.0000
3) -0.0508 0.0585 0.0599 -0.1812 0.1812 -0.1213 0.9045
4) -0.0211 0.0421 0.0941 -0.1515 0.1515 -0.0574 0.6235
ACKNOWLEDGMENT
torque for positive and negative direction. Consequently, the absolute value of the maximum required torque is reduced when the gravity compensation is applied. Next, the results when the robot moves in the air are evaluated. This movement is assumed to simulate the swing phase of the legs. Fig. 7 (b) shows that angle errors become slightly larger when a gravity compensation is applied. This result indicates that the gravity compensation may act as a disturbance for the joint control in the swing phase. Using Table V, the effectiveness of the gravity compensation mechanism is evaluated quantitatively. Comparing condition 1) with condition 2), the absolute average, the absolute max and the total power are reduced to 56%, 72% and 43% respectively by the gravity compensation. The absolute average and the total power for condition 3) is smaller about 10% than these indexes at condition 2). This result shows that the negative effects of the mechanism in the swing phase is not large. As described in the beginning of this chapter, the head and arms are still under development. Therefore, the total weight of the upper body is currently only 60% of the expected weight during the designing stage. As a result, in the present state the compensation ratio is at 0.83 ≈ 0.5/0.6, larger than the expected ratio of 0.5. Therefore, as is shown in Fig. 7 (a), the compensated torque during steady state rises to 83%, causing the joint angle error to be close to 0 ( ◦ ). The gravity compensation mechanism just gives a bias torque to joints, and thus, the effect is not affected by a speed of movement in theory. However, the influence of the motion speed to the gravity compensation must be carefully investigated in future. In the experiments, the joint angle errors are used to evaluate the effect of the gravity compensation mechanism. This method is insufficient for a quantitative evaluation. Therefore, the electric current of the motor will be used for the evaluation in the future work.
This research was supported by NEDO Industrial Technology Research Grant Program (project ID: 05A30703a). R EFERENCES [1] K. Nagasaka, Y. Kuroki, S. Suzuki, Y. Itoh, and J. Yamaguchi, “Integrated motion control for walking, jumping and running on a small biped entertainment robot,” in Proc. of IEEE Int. Conf. on Robotics and Automation, New Orleans, USA, April 2004, pp. 3189– 3194. [2] T. Nagasaki, S. Kajita, K. Kaneko, K. Yokoi, and K. Tanie, “A runnig experiment of humanoid biped,” in Proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Sendai, Japan, September 2004, pp. 136–141. [3] “http://world.honda.com/asimo/.” [4] S. Nakaoka, A. Nakazawa, F. Kanahiro, K. Kaneko, M. Morisawa, and K. Ikeuchi, “Task model of lower body motion for a biped humanoid robot to imitate human dances,” in Proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Edmonton, Alberta, Canada, August 2005, pp. 2769–2774. [5] M. Uchiyama, A. Konno, D. Sato, T. Matsumoto, Y. Ishida, and T. Ishihara, “A humanoid robot to perform tasks in which impact dynamics is utilized,” in Proc. of 36th Int. Symp. on Robotics, Tokyo, Japan, November 2005, p. TH1C1. [6] S. Shirata, A. Konno, and M. Uchiyama, “Design and development of a light-weight biped humanoid robot saika-4,” in Proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Sendai, Japan, September 2004, pp. 148–153. [7] K. Kaneko, F. Kanehiro, S. kajita, H. Hirukawa, T. Kawasaki, M. Hurata, K. Akachi, and T. Isozumi, “Humanoid robot hrp-2,” in Proc. of IEEE Int. Conf. on Robotics and Automation, New Orleans, USA, April 2004, pp. 1083–1090. [8] H. Hirukawa, F. Kanehiro, and S. Kajita, Open HRP: Open Architecture Humanoid Robotics Platform, R. A. Jarvis and A. Zelinsky, Eds. Springer-Verlag, 2003. [9] S. Hirose and R. Chu, “Development of a lightweight torque limiting m-drive actuator for hyper-redundant manipulator float arm,” in Proc. of IEEE Int. Conf. on Robotics and Automation, Detroit, USA, May 1999, pp. 2831–2836. [10] T. Morita, F. Kuribara, Y. Shiozawa, and S. Sugano, “A novel mechanism design for gravity compensation in three dimensional space,” in Proc. of IEEE/ASME Int. Conf. on Advanced Intelligent Mechatronics, Paper No. 280, 2003. [11] Y. Sugahara, Y. Mikuriya, T. Hosobata, H. Sunazuka, M. Kawase, K. Hashimoto, H. ok Lim, and A. Takanishi, “Support torque reduction mechanism for biped locomotor with parallel mechanism,” in Proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Sendai, Japan, September 2004, pp. 3213–3218.
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