Inclination Estimation and Balance of Robot Using

2010 IEEE-RAS International Conference on Humanoid Robots Nashville, TN, USA, December 6-8, 2010

Inclination Estimation and Balance of Robot using Vestibular Dynamic Inclinometer Vishesh Vikas and Carl D. Crane III Abstract— Traditionally, the sensor design for Inertial Sensing of Human movement/posture is done using an accelerometer-gyroscope combination while detection of static or dynamic activities is done using uniaxial accelerometers. A single axis accelerometer-gyroscope combination uses sensor fusion along with a Kalman Filter to estimate the inclination. The inclination is obtained by integrating angular velocity and acceleration combinations. Due to imperfect sensors, this leads to magnification of errors and drift over time. The sensor discussed in the paper, called the Vestibular Dynamic Inclinometer (VDI), uses two dual-axis accelerometers and one single axis gyroscope per axis to estimate the inclination. The concept of “equilibrium axis”, the axis along which the robot is at equilibrium, is discussed. The inclination angle obtained from the VDI is relative to the equilibrium axis and thus is more desirable as a control input rather than the inclination angle relative to the absolute gravity vector (as obtained from the accelerometer-gyroscope combination). The inclination angle obtained is independent of acceleration experienced by the robot and is not obtained by integration of angular velocity or acceleration unlike traditional designs.The control strategies proposed in the paper are torque control and acceleration control. Torque control requires generation of torque at the point of contact (analogous to the hip and ankle generating balancing torque in humans) and acceleration/postural control requires the acceleration of point of contact (analogous to running in humans) to maintain the body at equilibrium. The paper also proposes Lyapunov based non-linear adaptive controllers for an inverted pendulum (which approximates a human body) for both the control strategies. The controllers guarantee asymptotic stability. Simulations are performed assuming noise in the sensors.

II. P ROBLEM F ORMULATION

I. INTRODUCTION Sensor design for inertial sensing of human posture is done using an accelerometer-gyroscope combination [1], [2], [3]. Detection of static or dynamic activities can be done using uniaxial accelerometers [4]. Gyroscope-free designs using only accelerometers have been explored to detect inertial sensing [5], [6]. A single axis accelerometer-gyroscope combination uses sensor fusion along with a Kalman Filter to estimate the inclination. For all the methods, the inclination is obtained by integrating angular velocity and acceleration combinations. Due to imperfect sensors, this leads to magnification of errors and ”drift” over time. Robots from commercial companies (e.g. Sony, Honda) use gyroscopes and/or accelerometers in their active balancing, but details are not provided by the creators and quantitative comparison to humans are not available. The Vestibular Dynamic Inclinometer (VDI) has been introduced for static and dynamic [7], [8] Vishesh Vikas and Carl D. Crane III are with Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 326116250 {vishesh.vikas,carl.crane}@gmail.com

978-1-4244-8689-2/10/$26.00 ©2010 IEEE

inclination measurement of robots. The Vestibular Dynamic Inclinometer (VDI) is a novel design that uses two dualaxis accelerometers per axis to estimate the inclination. The design takes motivation from the human vestibular system [9]. The design is novel as it obtains the angular inclination directly from the sensor readings without integrating the angular velocity information. It outputs dynamic inclination of a body without requiring a significant computational burden, irrespective of gravity or sensor array location, and is not subject to drift or integration errors [8]. The paper discusses the Vestibular Dynamic Inclinometer(VDI) sensor and the concept of the Dynamic Equilibrium Axis(DEA), the axis along which the robot is at equilibrium. The inclination angle obtained from the VDI is relative to the DEA, and thus is more desirable as a control input rather than the inclination angle relative to the absolute gravity vector (as obtained from other inertial units). The inclination angle obtained is independent of acceleration experienced by the robot. The paper also proposes two control strategies - torque (analogous to the hip and ankle generating a balancing torque in humans) and acceleration/postural(analogous to running in humans) control are presented. Both are Lyapunov based non-linear adaptive controllers which guarantee asymptotic stability. Simulations are performed in MATLAB. The sensor noise is modeled as additive stationary white noise which is constant throughout the frequency spectrum.

A robot is modeled as an inverted pendulum as given in Figure 1. It is desired to sense the inclination of the robot and control it from falling over. The rigid body is modeled as a rod with mass m, center of mass C along the rod at a distance rC from the base of the rod, moment of inertia ICB at point C and angular damping coefficient Kd . Let point O be the base of rigid body in contact with a platform. Let N represent the inertial reference frame and B represent the reference frame fixed on the rigid body. Let g be the gravitational acceleration on the body, a be the acceleration ˜ of point O with respect to the earth reference frame and g be the resultant of the previous two mentioned accelerations. Thus ˜ =g+a g (1) A coordinate system fixed in the inertial reference frame with origin at O, Y-axis parallel to the ground, Z-axis into the plane of the paper, and fixed in reference frame N is defined with {ˆ x, yˆ, zˆ} as the orthonormal basis. Another coordinate system with origin O fixed in the body reference

245

B

B

L G P

R

G

L

C

R

P d/2

rC l X

Fig. 2.

g

N

a

Y O

Fig. 1. Model of robot with labeled coordinate systems in different reference frames

frame B, with {ˆ er , eˆθ , eˆz } orthogonal basis vectors in the radial, tangential directions, and into the plane of the paper respectively is defined. The Dynamic Equilibrium Coordinate system is defined to be fixed in the inertial reference frame N with origin at point O. The vectors {ˆ ex , eˆy , eˆz } form a set of orthonormal basis vectors such that eˆx is parallel to vector ˜ and eˆz is into the plane of the paper. Let φ be the angle g ˜ in clockwise direction. Let θ be the angle between g and g between g˜ and rigid body, as shown in Figure 1. The angular velocity between coordinate system N and B is denoted by N B ω which may be written as N

ω

B

˙ ez = θˆ

(2)

Point P is at a distance l radially along the body, whereas, the accelerometers R, L are located at a distance d/2 on either side of point P in a tangential direction of the body as shown in Figure 2. Let N aL/R denote the acceleration sensed by the L/R accelerometer.

Detail view of sensor - Accelerometers R, L and gyroscope G

[9]. A dual-axis accelerometer (MEMS) is assumed to be analogous to the otolith organs. Each vestibular organ is assumed to be analogous to a dual-axis accelerometer and a single-axis gyroscope. Human ears are placed symmetrically about the axis of symmetry (along which the nose lies), thus, providing motivation to design a sensor that has the vestibular-analogous accelerometer-gyroscope symmetrically placed about a symmetrical line. The gyroscope readings for both the gyroscopes will be theoretically the same (as they are attached to the same rigid body). However, the accelerometer readings for both accelerometers will be different. The design in Figure 2 is proposed which has two dual-axis accelerometers (L, R) symmetrically placed across a vertical line and one single-axis gyroscope (G). This sensor is called the Vestibular Dynamic Inclinometer(VDI). It has been shown that integration of angular acceleration to obtain angular velocity is prone to drift [8], thus, the use of the gyroscope to obtain the angular velocity is justified. B. Dynamic Equilibrium Axis At any point M on the body, performing a kinematic analysis [11] to obtain acceleration (N aM ) yields
N aM = N aO + N αB × rO→M + N ω B × N ω B × rO→M (3) where M may be {L, R, C}. It is known that

III. V ESTIBULAR DYNAMIC I NCLINOMETER

rO→C

=

rO→L/R

=

N

=

A. Sensor Design For measurement of the spatial orientation of the body, humans possess vestibular organs, the equivalent biological inertial measurement unit. They consist of two main receptor systems for inertial sensing - Semicircular canals (ducts) and Otolith organs. The semicircular canals are filled with viscous fluid, endolymph, which moves when the human body experiences angular acceleration in low frequency rotation and angular velocity in the mid- to high-frequency range [10]. This part is assumed to be analogous to a gyroscope(MEMS) sensor. The otolith organs are comprised of the utricle and saccule of the inner ear. The utricle is sensitive to a change in horizontal linear acceleration and the saccule is sensitive to the vertical linear acceleration

d/2

N

aO

rC eˆr d lˆ er ∓ eˆθ 2 ˜ g = g˜eˆx = −˜ g cos θˆ er + g˜ sin θˆ eθ

where aO denotes the acceleration sensed by the celerometer when it is placed at point O. Therefore,   N ¨eθ − θ˙2 eˆr aC = g ˜ + rC θˆ   d¨ N 2 ˙ aL = − lθ − θ − g˜ cos θ eˆr 2   d 2 ˙ ¨ + lθ + θ − g˜ sin θ eˆθ 2   d N aR = − lθ˙ 2 + θ¨ − g˜ cos θ eˆr 2   d 2 ¨ ˙ + lθ − θ − g˜ sin θ eˆθ . 2

246

(4) ac(5)

(6)

(7)

Let the reaction forces at point O be FR . Applying Newton’s second law and Euler’s second law about the center of mass C gives   ¨eθ − θ˙2 eˆr ) = FR . m ˜ g + rC (θˆ (8) ˙ ez + (−rC eˆr ) × FR . ICB · N αC = −Kdθˆ

Thus,


2 ¨ ICB + mrC g rC sin θ θ = −Kd θ˙ + m˜ ¨∗

(9) (10)

C. Mathematical Manipulations In Equations 6 and 7 the kinematic analysis of the body is performed. It is important to observe that all the calculations are independent of the dynamics of the body, i.e. control torque, external force, etc. Four readings are obtained from the two accelerometers (radial and tangential components in Equations 6, 7). Calculating the difference and mean of the readings and calling them ζ1 , ζ2 yields =

F

aL − F aR

(11)


1 F aL + F aR ζ2 = 2 ζ1 = |{z} dθ¨ eˆr + |{z} dθ˙2 eˆθ . ζ1r

(12) (13)

ζ1θ

    ζ2 = − lθ˙2 − g˜ cos θ eˆr + lθ¨ − g˜ sin θ eˆθ {z } {z } | | ζ2r

(14)

The control strategy to bring the body to equilibrium can be torque application (analogous to torque on human body for equilibrium via hip, ankle) or acceleration/de-acceleration of the body (analogous to leaning of human body while accelerating or de-accelerating). The former is referred to as torque control and the latter as acceleration control or postural throttling. It is desired to design a controller for varying weight, unknown moment of inertia, and damping coefficient. For these reasons, Lyaponov based non-linear controllers are proposed. It is desired to bring the rigid body back to equilibrium. To achieve the control purpose, a tracking error e ∈ R is defined as e , θd − θ = −θ

l d ζ1r

− ζ2θ = sin θ

l d ζ1θ

+ ζ2r . cos θ

(20)

where the desired angle of inclination θd ∈ R is zero for all time. For stability analysis of the system, filtered tracking error r ∈ R is defined as r , e˙ + αe = −θ˙ − αθ

(21)

where α ∈ R is a positive real constant. A. Torque Control In this case, the torque is applied on the body and the equations of motion change from Equation 10 to

B where IO 1×3 Y ∈R

B¨ g rC sin θ = TC (22) IO θ + Kd θ˙ + m˜
B 2 = IC + mrC and TC is the control torque. is defined as h i ˙ g˜ sin θ, −αθ˙ . Y , θ, (23)

The control torque is designed as ˆ − kr TC = Y Θ

ζ2θ

Thus, the angular acceleration, velocity and inclination can be obtained as ζ1r (15) θ¨ = d l ζ1r − ζ2θ sin θ = d (16) g l ζ1θ + ζ2r cos θ = d (17) g l ζ1r − ζ2θ (18) tan θ = dl d ζ1θ + ζ2r g˜ =

IV. C ONTROL S TRATEGY

˙∗

From Equation 10, the equilibrium position (θ = θ = 0) of the system is θ∗ = 0. The aim of the problem is to bring the body to its equilibrium position which is no more the absolute vertical position (i.e. direction parallel to the gravity vector g). The new equilibrium position is defined as the Dynamic Equilibrium Axis (DAE) which is parallel to the ˜ , inclined at angle φ to g. resulting acceleration on the body g The Dynamic Equilibrium Axis is dependent on the resultant acceleration acting on the body, and thus is time-varying (more precisely, acceleration varying). When the body is accelerating, φ is positive, when its de-accelerating, φ is negative. This fact can be observed when humans lean forward (change equilibrium axis) when trying to accelerate (sprint) and bend backwards while attempting to de-accelerate. Both the cases display how the equilibrium axis (DEA) changes when acceleration is experienced by the body.

ζ1

It is important to observe that tan θ (Equation 18) is independent of the resultant acceleration g˜ acting on the body, but, can be uniquely determined (assuming g˜ > 0). It is also possible to determine the resulting acceleration magnitude g˜ ˙ is directly observed from Equation 19. Angular velocity (θ) from the gyroscope (G) reading. Obtaining angular velocity from integration of angular acceleration (Equation 15) or from Equation 11 is prone to drift and undesirable error [8].

(24)

where k ∈ R is a positive constant and the update law for ˆ ∈ R3×1 is Θ ˆ˙ = ΓY T r Θ (25) where Γ ∈ R3×3 positive definite adaptation gain matrix and α, k are constrained as follows 1 (26) α, k > . 2 Theorem 1: The controller given in Equations 24 and 25 with gain conditions given in Equation 26 ensures that the tracking error is regulated as follows

(19)

247

lim ||e(t)|| = 0,

t→∞

lim ||e(t)|| ˙ =0

t→∞

thus, assuring global asymptotic stability. Proof: Differentiating the filtered tracking error yields ˙ r˙ = e¨ + αe˙ = −θ¨ − αθ. Multiplying Equation 27 by tion 22

B IO

(27)

and simplifying using Equa-

B ˙ g r sin θ − IO αθ − TC = Kd θ˙ + m˜

B IO r˙ B IO r˙

= Y Θ − TC

where Θ ∈ R as

3×1

B

C

(28) (29)

is unknown, yet deterministic and defined   B T . Θ , Kd , m, IO

(30)

˜ = Θ − Θ. ˆ Θ

(31)

˜ ∈ R3×1 is defined as Θ

g X

N

FD

Combining Equation 24, 29, 31 gives B IO r˙

˜ − kr. =YΘ

1 B 2 1 2 1 ˜ T −1 ˜ I r + e + Θ Γ Θ. 2 O 2 2 Differentiating 33 and simplifying yields V ,

(33)

˜˙ ˜ +Θ ˜ T Γ−1 Θ V˙ = −kr2 − αe2 + er + Y Θr

(34)

using the Arithmetic Mean-Geometric Mean inequality r 2 + e2 . 2

(35)

ˆ˙ Simplify˜˙ = −Θ. Θ is unknown and deterministic. Thus, Θ ing Equation 34 using 35 and 25 gives     1 2 1 2 r − α− e . (36) V˙ ≤ − k − 2 2 Using the constraints given in Equation 26, the expression in Equation 36 is upper bounded by a continuous, negative semi-definite function. By using Barbalat’s Lemma [12] lim ||e(t)|| = 0,

t→∞

lim ||e(t)|| ˙ =0

t→∞

∀θ ∈ R. (37)

B. Acceleration Control or Postural Throttle In this case, the control and motion looks as shown in Figure 3. The control of the body is a force in a specific direction (ˆ y), and thus it is safe to assume that the acceleration in the other direction (ˆ x) is known/calibrated (usually gravitational). Thus, ˆ. g˜ = gˆ x + ah y

(38)

Let β = (θ + φ), be the inclination of the body relative to the absolute gravity g. Now, the equations of motion change to mrC cos βah + Kd θ˙ − mrC sin βg mrC cos β θ¨ + mah − mrC sin β θ˙2 − FC

B¨ θ+ IO

FC

(32)

Let V denote a continuously differentiable positive definite radially unbounded Lyapunov function candidate defined as

er ≤

ah

O

Y

Fig. 3. Motion in known direction as control parameter. Here, β = φ + θ, inclination of the body from the absolute gravity g.

  g = cos (41) φ = tan g˜ p where FC is the control force and g˜ = g 2 + a2h . Defining Y ∈ R1×4 as h i    ˙ −αθ, ˙ −gsβ , θ˙ + r cβ − αsβ sβ θ˙ . Y , θ, (42) −1



ah g



−1

The control force is designed as  1  ˆ Y Θ + kr FC = − cos β

(43)

where k ∈ R is a positive constant and the update law for ˆ ∈ R4×1 is Θ ˆ˙ = ΓY T r Θ (44) where Γ ∈ R4×4 is a positive definite adaptation gain matrix and α, k are constrained as follows 1 α, k > . (45) 2 It is worthwhile to mention that the system becomes uncontrollable when β = π/2 i.e. when the rigid body is parallel to the direction of acceleration. Therefore, cos β > 0 for all controllable cases. Theorem 2: The controller given in Equations 43 and 44 with gain conditions given in Equation 45 ensures that the tracking error is regulated as follows lim ||e(t)|| = 0,

t→∞

lim ||e(t)|| ˙ =0

t→∞

thus, assuring global asymptotic stability. Proof: The proof proceeds on similar lines as the proof for the previous theorem. Solving Equations 39, 40 for θ¨ and ah gives

= 0 (39) = 0 (40)

248

Aθ¨ = −Kd θ˙ − mrc2 sβ cβ θ˙2 + mgrc sβ − rC cβ FC Aah =

B IO

m

(46)

2 B sβ cβ g (47) FC + IO rC sβ θ˙2 + Kd rC cβ θ˙ − mrC

  2 2 where A = ICB + mrC sβ . It can be observed that A > 0∀β. Calculating the following expression using the expression for the filtered tracking error as defined in Equation 21 gives

˙ Ar˙ + Ar = Y Θ + cβ FC rC

θ(0) = 7 o

7

(48)

o

θ(0) = 9

o

θ(0) = 10 o

6 5 4 3 2

0

˜ = Θ − Θ. ˆ Θ

(50)

(51)

Let V denote a continuously differentiable positive definite radially unbounded Lyapunov function candidate defined as 1 1 1 ˜ T −1 ˜ Ar2 + e2 + Θ Γ Θ. 2rC 2 2

(52)

Differentiating 52, using Equation 51 and simplifying yields ˜˙ ˜ +Θ ˜ T Γ−1 Θ. V˙ = −kr2 − αe2 + er + Y Θr

(53)

˜˙ = −Θ. ˆ˙ Simplifying As Θ is unknown and deterministic, Θ Equation 53 using 35 and 44 gives     1 2 1 2 ˙ r − α− e . (54) V ≤− k− 2 2 Using the constraints given in Equation 45, the expression in Equation 54 is upper bounded a by continuous, negative semi-definite function. By using Barbalat’s Lemma [12] lim ||e(t)|| = 0,

θ(0) = 8

1

˙ Ar˙ + Ar ˜ − kr. = YΘ rC

t→∞

Value 2m 1.8 m mL2 /3 9.81 m/s2

Angle Response over time

8

−1 0

Combining Equations 43, 48, 50 gives

V ,

Parameter l L IO g

PARAMETERS OF SYSTEM USED FOR SIMULATION

where Θ ∈ R4×1 is unknown, yet deterministic and is defined as T  Kd ICB (49) , , m, mrC . Θ= rC rC ˜ ∈ R4×1 is defined as Θ

Value 85 kg 0.25 m 1.25 m 2 N m s rad−1 TABLE I

Kd ˙ ICB ˙ = αθ − mgsβ + cβ FC θ− rC rC    +mrC θ˙ + r cβ − αsβ sβ θ˙

Angle (degrees)

˙ Ar˙ + Ar rC

Parameter m d R Kd

lim ||e(t)|| ˙ =0

t→∞

∀θ ∈ R. (55)

; V. SIMULATION R Simulations were performed in MATLAB . The sensor noise was modeled as stationary white noise which is constant throughout the frequency spectrum based on the specification data sheet √ of accelerometer ADXL213 (noise density of 160 µg/ Hz rms)√and gyroscope ADXRS450 (noise density of 0.015 o /sec/ Hz) manufactured by Analog Devices. The frequency of operation of the sensors is assumed to be 10 Hz. Values of other parameters are motivated by modeling of the human body as an inverted pendulum. The values for simulation are displayed in Table I. For the simulation of the torque control strategy, the gains are assumed to be α = 4, k = 2, Γ = 103 I3×3 . The

1

2

3

4

5 6 Time (sec)

7

8

9

10

Fig. 4. Angular Response for different initial inclinations, zero initial ˙ velocity(θ(0) = 0) and torque control strategy

learning of parameters is done online. Angular response and Control Torque comparisons for different initial inclinations are shown in Figure 4 and 5 respectively. It is important to observe in Figure 4 that the rigid body balances to ±0.5o in a relatively quick time and remains in that vicinity thereafter. Also, the torque required to maintain the rigid body at that inclination is relatively small when the foot is modeled as a point. In real life, the foot is not a point of contact, rather a surface of contact, thus assuring better balance. The peak torque generated is a little high but the results are very encouraging. For the simulation of the acceleration control strategy, the gains are assumed as α = 1, k = 4, Γ = 25I4×4 . Angular response (β, not θ as β is the absolute position of the body), Control Force and horizontal acceleration comparisons for different initial inclination are shown in Figures 6, 7 and 8 respectively. The horizontal acceleration increases accordingly to balance the robot at the desired DEA. The peak control force seems higher than normal (gait forces for a 85 kg human), but, the transient response looks good in a relatively large neighborhood. VI. CONCLUSIONS AND FUTURE WORKS It has been shown that the equilibrium position (Dynamic Equilibrium Axis), changes with the acceleration being experienced by the robot. The Vestibular Dynamic Inclinometer (VDI) returns the inclination to the DEA. This is more helpful as a control input as the goal is to bring the rigid body to a state of equilibrium i.e. align it along the DEA. The sensor works in varying acceleration conditions e.g. earth gravity, accelerating elevator, accelerating ship, or moon gravity. It is capable of sensing the magnitude of the acceleration acting on the body.

249

Control Torque over time

800

θ(0) = 8 o

600

θ(0) = 9

Control Force over time

1800

θ(0) = 7 o

θ(0) = 15 o θ(0) = 20 o

1600

o

θ(0) = 25

θ(0) = 10 o

o

θ(0) = 30 o

1400 1200 Force (N)

Torque (Nm)

400 200 0

1000 800 600

−200

400 −400

200

−600 0

1

2

3

4 5 Time (sec)

6

7

8

9

0

10

0

Fig. 5. Control Torque for different initial inclinations, zero initial ˙ velocity(θ(0) = 0) and torque control strategy

42 40

θ(0) = 20 θ(0) = 25

o

θ(0) = 30

o

6

8

10 12 Time (sec)

14

16

18

20

Horizontal Acceleration over time

30

θ(0) = 15 o o

4

Fig. 7. Control Force for different initial inclinations, zero initial ˙ velocity(θ(0) = 0) and acceleration control strategy

β Response over time

44

2

θ(0) = 15 o

25

θ(0) = 20

o

θ(0) = 25

o

θ(0) = 30 o

a (m/sec2) h

β(degrees)

20

38

15

36 34

10

32

5

30 0

28 26 0

2

4

6

8

10 12 Time (sec)

14

16

18

−5 0

20

2

4

6

8 10 12 Time (sec)

14

16

18

20

Fig. 6. Angular Response(β) for different initial inclinations, zero initial ˙ velocity(θ(0) = 0) and acceleration control strategy

Fig. 8. Horizontal Acceleration(ah ) response for different initial inclina˙ tions, zero initial velocity(θ(0) = 0) and acceleration control strategy

Two control strategies - torque and acceleration either bring the body back to the DEA or change the DEA. The non-linear adaptive controllers for the strategies theoretically assure global asymptotic stability. During simulation, the sensors were modeled with Gaussian white noise constant throughout the frequency spectrum. In case of torque control, the response of the system is good and the equilibrium maintenance torque is relatively low assuming a point contact model for a surface contact foot. The response for acceleration control is very encouraging. The controllers work in varying acceleration conditions (because of the sensor output). In the near future, it is desired to explore the possibility of parameter estimation through learning control. This is motivated to learn from previous experiments and their results on the same robot. The tuning of the gains to obtain desired quick transient response (quick learning) in a large neighborhood of operation (e.g. ±45o ) should be done. Hybrid controller of both control strategies should be developed to achieve desired robot trajectory.

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R EFERENCES [1] H. Luinge, “Inertial sensing of human movement,” Unpublished PhD, University of Twente, Enschede, the Netherlands, 2002. [2] H. Luinge and P. Veltink, “Measuring orientation of human body segments using miniature gyroscopes and accelerometers,” Medical

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