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Bioinspired dynamic inclination measurement using inertial sensors

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Bioinspir. Biomim. 10 (2015) 036003

doi:10.1088/1748-3190/10/3/036003

PAPER

Bioinspired dynamic inclination measurement using inertial sensors RECEIVED

7 August 2014 REVISED

8 January 2015 ACCEPTED FOR PUBLICATION

28 February 2015

Vishesh Vikas1 and Carl Crane Center of Intelligence Machines and Robotics, University of Florida, Gainesville, FL 32611 USA 1 Author to whom any correspondence should be addressed. E-mail: [email protected] and [email protected]

PUBLISHED

16 April 2015

Keywords: inertial sensing, dynamic orientation, dynamic equilibrium, bioinspiration, vestibular system Supplementary material for this article is available online

Abstract Biologically, the vestibular feedback is critical to the ability of human body to balance in different conditions. This balancing ability inspires analysis of the reference equilibrium position in dynamic environments. The research proposes and experimentally validates the concept of equilibrium for the human body modeled as an inverted pendulum, which is instrumental in explaining why we align the body along the surface normal when standing on a surface but not on an incline, and tend to lean backward or forward on non-static surfaces e.g. accelerating or decelerating bus. This equilibrium position—the dynamic equilibrium axis—is dependent only on the acceleration of surface of contact (e.g. gravity) and acts as the reference to the orientation measurements. The research also draws design inspiration from the two human ears—symmetry and plurality of inertial sensors. The vestibular dynamic inclinometer and planar vestibular dynamic inclinometer consist of multiple (two or four) symmetrically placed accelerometers and a gyroscope. The sensors measure the angular acceleration and absolute orientation, not the change in orientation, from the reference equilibrium position and are successful in separating gravity from motion for objects moving on ground. The measurement algorithm is an analytical solution that is not time-recursive, independent of body dynamics and devoid of integration errors. The experimental results for the two sensor combinations validate the theoretically (kinematics) derived analytical solution of the measurement algorithm.

1. Introduction Mechanically, the human body displays a remarkable quality of maintaining equilibrium for a body that is in a state of unstable equilibrium (biped stance). Biologically, this balancing ability requires both visual and vestibular feedback. A well known cause of instability is failure of vestibular sensors [2] and the body enjoys stability even during lack of visual feedback. The human vestibular organs, biological equivalent of inertial measurement units (IMUs), measure spatial orientation of the body. These vestibular organs lie in the inner ear and are affected only by force fields such as gravity and acceleration [3, 4]. They consist of two main receptor systems [5] for inertial sensing— semicircular canals(ducts) and otolith organs, as shown in figure 1. The semicircular canals are the primary systems responsible for sensing angular motion of the head and transmitting the information to the brain stem. Each canal is comprised of a circular © 2015 IOP Publishing Ltd

tube containing fluid continuity, interrupted at the ampulla (that contains the sensory epithelium) by a water tight, elastic membrane called the cupula. The three semicircular canals, arranged in orthogonal planes, take advantage of endolymph fluid dynamics to sense angular motion [6]. Due to mechanical factors (such as fluid adhesion), the canals code angular acceleration during low frequency rotation and angular velocity in the mid- to high-frequency range [3]. The otolith organs comprise of utricle and saccule. They are two sac-like structures each of which contains a specialized region, the macula, made up of a ciliated sensory epithelium (the vestibular hair cells) that are sensitive to direction and magnitude of linear acceleration [7]. Collectively, both these organs provide the vestibular system with the ability to measure angular velocity and linear acceleration . Inertial sensors, accelerometers and gyroscopes, measure linear acceleration and angular velocity respectively. Hence, each vestibular organ can be

Bioinspir. Biomim. 10 (2015) 036003

V Vikas and C Crane

Figure 1. The vestibular system is shown as the red drawing with the three semicircular canals that sense rotational movements, and the utricle and saccule that are sensitive to linear acceleration. PC, SC, and HC are the posterior, superior, and horizontal canals, respectively (Reproduced with permission from [1]).

Figure 2. The human body modeled as an inverted pendulum with center of mass at point C, the inverted pendulum can rotate and translate on a surface where point O is the point of contact (zero relative acceleration) between the surface and the body. The resultant acceleration of point O, g˜ , is the sum of gravitational acceleration g and non-gravity acceleration a . Coordinate systems N , D and B are defined by basis {X , Y , Z }, {E1, E 2, E 3} and {e1, e 2, e 3}, respectively. The unit vectors Z , E 3, e 3 are parallel to g , g˜, rO → C , respectively. The body is in equilibrium when it aligns itself along the dynamic equilibrium axis (DEA)—axis parallel to g˜ .

assumed to be analogous to an accelerometer and a gyroscope. Human ears are placed symmetrically about the axis of symmetry (along which the nose lies), thus, providing inspiration to design a sensor that has vestibular-analogous accelerometer and gyroscope combination placed symmetrically about an axis. Analysis of gyroscope readings indicates that, theoretically, both the readings are identical as they are attached to the same rigid body. However, the accelerometer readings for the two accelerometers will be different. Throughout the literature, the human vestibular system-sensor analogy is drawn as a single gyroscope and single accelerometer. To calculate orientation, the 2

sensor signals are fused and orientation is estimated using time-recursive Kalman filter-like algorithms. As the linear accelerations in each of the vestibular organs are different, such analogy leads to loss of critical information. The presented research shows this additional information facilitates the bio-inspired inertial sensors to instantaneously measure orientation based on single measurement alone without need for timerecursive algorithms. The measurement of spatial orientation is traditionally performed using accelerometer(s), gyroscope or accelerometer-gyroscope combination. Accelerometers sense linear acceleration and can measure

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orientation only when non-gravitational acceleration is negligible [8] due to their inability to differentiate between motion and gravity [9]. On the other hand, gyroscopes measure the angular velocity and the relative change in orientation can be calculated by integrating the angular velocity using strapdown integration algorithms [10–12]. However, integration of the noise in measurement of angular velocity results in cumulative integration errors. This problem can be addressed either by using gyroscope-free all-accelerometer inertial navigation techniques or combining accelerometer and gyroscope signals. Gyroscope-free all-accelerometer inertial navigation techniques have been explored using three (with limitations on angular velocity), six [13], nine [14, 15] or twelve [16] strategically placed accelerometers. Another approach to tackle the errors of the cumulative errors involves combining both gyroscope and accelerometer signals. Inertial Measurement Units (IMUs) typically comprise of an accelerometer and a gyroscope [17, 18] and fuse the sensor data using Kalman filter like estimation algorithms with knowledge of (error) dynamics of the system [19–23]. Another quantity of interest along with orientation is the angular acceleration. Angular acceleration can be exclusively determined using three non-collinear points on a rigid body given the angular velocity measurements [24], filtering techniques [25] and multiple (twelve or nine) accelerometer combination [26]. The research presents designs of two novel bio-inspired sensors—the Vestibular Dynamic Inclinometer (VDI) and the planar Vestibular Dynamic Inclinometer (pVDI)—that comprise of symmetrically placed accelerometers and a gyroscope. These sensors simultaneously measure the orientation parameters—orientation angle(s), angular velocity and also the angular acceleration. The measurement algorithm consists of an analytical solution that does not require integration of the gyroscope signal or knowledge of body dynamics and is independent of gravity, more specifically, the acceleration of surface of contact. The efficacy of the sensors comes from the fact that they are simple, require low computation, not time-recursive and provide the dynamic orientation without integrating the angular velocity and applying any time-recursive filtering techniques for sensor fusion. The angular calculations are valid for large angles, are independent of the acceleration of the surface of contact (gravity, etc) and are also independent of dynamics of the body. Interestingly, the vestibular feedback facilitates for human postural adjustment in the cases when surface of contact is non-static e.g. in an elevator, on a moving bus, while sprinting, etc. This observation raises the question—how is the equilibrium position defined? Is there a relationship between non-static nature of surface of contact and equilibrium position? For humanoid and biped locomotion purpose, the concept of the zero moment point (ZMP) [27] is extensively used for equilibrium analysis. The ZMP plays an important 3

role for gait analysis, synthesis, and control. However, while analyzing a human simplified as an inverted pendulum, it is observed that the equilibrium position of the body changes with non-static nature of the surface of contact and is only dependent on the acceleration of surface of contact. This concept of the dynamic equilibrium axis (DEA) related to the dynamic equilibrium of rigid body as discussed in section 2.2. The DEA acts as an absolute reference to orientation measurements from the VDI and the pVDI sensors. The paper is structured as three more sections and concluding remarks—the next section defines the problem, discusses dynamic equilibrium and the DEA. The following section analyzes the problem for the planar case and presents the bio-inspired VDI. The third section extends the VDI sensor to pVDI which is utilized to tackle the problem of the body capable of moving on a surface. Both sections present experimental results for respective sensor setup.

2. Problem formulation and dynamic equilibrium 2.1. Problem formulation The human body is modeled as a simplified inverted pendulum capable of translating and rotating with point O as the instantaneous point of contact between the rigid body and the surface as shown in figure 2. The inverted pendulum has mass m with the center of mass located at point C, moment of inertia IBC at point C and angular damping coefficient Kd. Let N, B represent coordinate systems fixed in inertial and body reference frames respectively and origin at point O. Coordinate systems N, B are defined by the orthonormal basis {X , Y , Z }, {e1, e 2, e 3} respectively where Z is normal to the surface of contact and e 3 parallel to the vector joining points O and C. Let g be the gravitational acceleration (parallel to Z ). The pendulum can move on a surface as long as the relative linear acceleration between the surface and pendulum at point of contact O is zero. Let a be the non-gravity acceleration of point O with respect to (w.r.t.) the inertial reference frame and g˜ be the resultant of point O. N

aO = BaO = g˜ = g + a ,

(1)

where I aM represents the acceleration of point M w.r. t. coordinate system I. The Dynamic Equilibrium coordinate system D is defined to be fixed in the inertial reference frame N with origin at point O and {E1, E 2, E 3} as the orthonormal basis such that E 3 is parallel to vector g˜ . The angular velocity and acceleration of coordinate system I w. r. t J are denoted by J ωI , J I α respectively. It is desired to measure the orientation parameters—the orientation angle(s), angular velocity and angular acceleration of the body.

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Figure 3. Equilibrium position for the human body modeled as a simplified inverted pendulum. Gravitational acceleration, nongravity dynamic acceleration and their resultant are denoted by g , a , g˜ , respectively. The dotted line represents the surface normal. While standing, the human body aligns itself along the gravity which is (a) coincident with the surface normal while standing on flat surfaces, but, (b) the same is not true while standing on an incline. The tendency of humans to (c) lean backward while traveling on an accelerating bus is an attempt to align the body along the new DEA parallel to g˜ . (d) The DEA changes when the bus decelerates. In cases of turning cars, non-gravitational the centrifugal acceleration a changes the DEA to the direction opposite to the turn—(e) left lean for a right turn and (f) right lean for a left turn.

2.2. Dynamic equilibrium The linear acceleration of any point M ( N aM ) located at some distance rO → M from point O on the rigid body can be written as N

for Λ =

aM = N aO + ΛrO → M

N B ∧

( α)

+

((

N

ωB

∧ 2

) ),

(2)

where the cross

product hat operator matrix [28] is defined such that for vectors x , y , (x)∧ y = x × y . To analyze for equilibrium position, Euler first and second law about C yields IBC · N α B = −K dN ωB − rO → C × m · N aO .

(3)

For rotational equilibrium N ωB* = 0 and N α B * = 0. Therefore, it can be concluded that the body is in the equilibrium position if rO → C is parallel to N aO . This new equilibrium position is defined as the Dynamic Equilibrium Axis (DEA) which is parallel to g˜ , the resulting acceleration of point of contact O. Hence, the DEA is only dependent on the resultant acceleration of the surface of contact i.e. point O, and is, thus, timevarying (more precisely, acceleration varying). This equilibrium axis changes the intuitive definition of equilibrium position being upright and normal to the surface of contact. This is only true while standing on flat surfaces (figure 3(a)) and changes when human body stands on an incline aligning itself to the gravitational acceleration, not the surface normal (figure 3(b)). It successfully explains the tendency of humans to slightly lean backward or forward when traveling on an accelerating and decelerating buses 4

(figures 3(c), (d)). Similarly, leaning left or right while being seated inside a car turning right or left (opposite direction) can be explained by desire to align the body along the resultant of gravitational and centrifugal acceleration (figures 3(e), (f)). Another very common observance of change in DEA is observed when humans lean forward when trying to accelerate (sprint) and bend backward while attempting to decelerate at the end of the run. The information from extra accelerometer facilitates orientation measurement from this new reference, the DEA, as the aim of the human body is to bring body to equilibrium. This is also observable while traveling on an elevator where the acceleration of the base platform varies the effective magnitude of the gravity acting on the body, however, the orientation measurement from the vestibular organs remains same as the direction of gravity remains unchanged. It should also be observed that when the body is not in contact with the ground and experiences free fall, the concept of the DEA ceases to exist as the acceleration experienced by the linear accelerometer at point O is zero, i.e., g˜ = 0. Theoretically, this reinforces the concept of the DEA as the idea of an ‘equilibrium position’ ceases to exist in zero gravity conditions. The analysis indicates that the rotational equilibrium position for the robot is not a point or a surface, but an axis—the DEA. Consequently, the orientation of the body about the DEA is irrelevant for robot equilibrium e.g. humans are at equilibrium while standing upright irrespective of facing East, West, North or South.

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Figure 4. (a) Model of a robot capable of planar motion with detailed view of the mounted (b) vestibular dynamic inclinometer (VDI) sensor. The VDI sensor takes design inspiration from the two symmetrically placed human ears—two symmetrically placed linear accelerometers (L, R) and a gyroscope (G).

3. Vestibular dynamic inclinometer

ζ2 =

3.1. Sensor design This section analyzes the problem stated in section 2.1 for the planar case as illustrated in figure 4(a) where the body is free to rotate about point O (revolute joint) while point O is free to translate in Y–Z plane. The VDI sensor, discussed next, is located at distance l along the body at point P. Let θ, ϕ, β be defined as follows ϕ = Z · e 3, θ = e 3 · E 3, β = ϕ + θ.

3.2. Orientation measurement—analytical solution The symmetric location of the VDI accelerometers implies d e 2, 2

rO → R = l e 3 +

d e2 2

(5)

( NaL − NaR ) = β˙ 2e d

2

− β¨e 3

(6)

5

⎜

⎝l

θ

⎞ + β¨⎟ e 2 ⎠ (7)

where sj = sin(j), cj = cos(j). The components of the vectors ζ i , will be referred to as ζij , j = 1, 2, 3. Using equations (6), (7), the angular acceleration, inclination and acceleration of point O can be obtained as β¨ = ζ13

(8)

) l ( ζ13 − ζ22 ) l ( ζ12 + ζ23 ) = g˜ = .

(9)

(

θ = atan2 ζ13 − ζ22, ζ12 + ζ23

sθ

cθ

(10)

It is important to observe that θ (equation (9)) can be uniquely determined as g˜ > 0 and is independent of the resultant acceleration of surface of contact (g˜ ). It is also possible to determine the resulting acceleration magnitude g˜ from equation (10). Obtaining the angular velocity ( β˙ ) is a little tricky, as the sensor mathematics is able to obtain the magnitude of the 2 angular velocity ( β˙ ), but not the direction. So, the angular velocity can be obtained by integration of the angular acceleration ( β˙I ), from accelerometers ( β˙A ) or from the gyroscope readings. β˙I =

ζ1, ζ2 denote the weighted difference and mean of two accelerometer readings. Using equation (5), they can be written as ζ1 =

2l ⎛ g˜ 2⎞ + ⎜ cθ − β˙ ⎟ e 3, ⎝l ⎠

(4)

So, N ωB = β˙ Z , N α B = β¨Z . It is desired to measure the inclination parameters—inclination angle (θ), angular velocity ( β˙ ) and angular acceleration ( β¨) independent of acceleration of surface of point O (g˜ ). The proposed design, as shown in figure 4(b), consists of 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) and is identified through point P.

rO → L = l e 3 −

( NaL + NaR ) = −⎛ g˜ s

∫ β¨dt ,

β˙A = sign

( ∫ β¨dt)

ζ12 .

(11)

Calculation of angular velocity as β˙I is prone to accumulation of error (drift), while, calculation of angular velocity as β˙A will result in magnification of contribution of noise near zero as a square root is involved [29]. Therefore, for this research, the reading

Bioinspir. Biomim. 10 (2015) 036003

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Figure 5. Experimental setup for validating the VDI sensor (figure 4) and the DEA. The accelerometers are symmetrically located at 25 cm about the inverted pendulum and the gyroscope is strapped onto the pendulum as indicated. The magnetic encoder is assumed to be the ground truth and is located at the base revolute joint. A linear accelerometer is located on the base to measure the acceleration of the base when moved.

from the gyroscope is used to measure the angular velocity. 3.3. Experiment and discussion The experiment was set with two dual-axis linear MEMS accelerometers (ADXL320 with noise density 1 of 250 μg Hz− 2 rms) symmetrically placed at d = 25 cm about the center-line of the inverted pendulum at distance l = 50 cm from the base as illustrated in figure 5. A single-axis MEMS gyroscope 1 (ADXRS613 with noise density of 0.04° s−1 Hz− 2 rms) was also strapped onto the inverted pendulum. The accelerometers were calibrated using the autocalibration algorithm [30] and the gyroscope drift was modeled as a first-order Markov process, driven by a small white Gaussian noise [22]. A linear MEMS accelerometer (ADXL320) was fixed to the base of the inverted pendulum to sense the acceleration of the base. Magnetic encoder (US Digital MA3-A10-125-N with noise density of 0.03° rms) was fixed at the revolute base joint. All the measurements were sampled at 50 Hz and the encoder angle readings were assumed to be the ground truth. The encoder angle measurements were differentiated to obtain the assumed true angular velocity and acceleration. The first experiment was performed keeping the base fixed, i.e., N aO = g E 3 (figure 4) to observe the VDI sensor measurements. The plots of comparison of orientation parameters are shown in figures 6(a), 7(a) and (b). The observations had standard error of 0.048 deg , 0.232 deg s−1, 1.672 deg s−2, 0.002 m s−2 while the error distribution had the standard deviation of 0.504 deg , 2.987 deg s−1, 19.873 deg s−2, 0.071 m s−2 for inclination angle, angular velocity, 6

angular acceleration and gravitational acceleration respectively. The gyroscope for the experiment saturated at ± 75 deg s−1. Closely observed, the angular acceleration measurements are very good while erring at peaks and the relative error is around 10%. The possible reasons for errors in measurement of angular acceleration are accuracy of placement of accelerometers (d , l ), calibration errors and noise (accuracy) in the inertial sensors. The instantaneous measurement of the angular acceleration along with other orientation parameters is unique to this sensor and is not obtained with traditional IMUs. Another consequence of this observation and equation (8) may be in formulation of estimation filter dynamics. Instantaneous angular velocity measurements from the gyroscopes were observed to be more accurate than those obtained via accelerometers through equation (11). This, however, does not mean that gyroscope-free VDI cannot be used for estimation of angular velocity. The equations(6), (7) are expected to be important in inclination parameter estimation in the equivalent gyro-free VDI sensor. The second experiment comprised of the previously described setup, but with accelerating base (a = ah Y ). The accelerating base simulates an accelerating platform such that the DEA (reference to the VDI sensor) is no more aligned to the vertical as in case of the previously discussed experiment with static base. The encoder readings provide inclination from the absolute vertical, which are sum of the VDI sensor readings (θ) and the DEA (ϕ) as stated in equation (4). The measure of absolute inclination angle β is compared with the encoder angle in figure 6(b) resulting in standard error of 0.092 deg and standard deviation of 0.415 deg . The angle ϕ of the DEA is calculated by assuming gravitational acceleration as 9.81 m s−2 and acceleration obtained from the base accelerometer (figure 5). This experiment is performed to analyze the concept of the DEA and experimental observation of ϕ (via β) illustrates the same. The results reiterate the hypothesis that equilibrium axis (ϕ) changes with change in acceleration point O (g˜ ). The inclination angle (θ) is the desired angle that will align the body along the DEA, the new reference position which changes over time. The DEA is aligned along the gravity (surface normal) when the base is static, therefore, in that case, the VDI is able to separate gravity from motion of the rotating body.

4. Planar vestibular dynamic inclinometer 4.1. Sensor design This section analyzes the case (section 2.1) where the body is free move on surface. Here, aligning the body along the equilibrium axis requires one to move it about an equivalent universal-joint (two rotational degrees-of-freedom). As discussed earlier, the orientation of the body about the DEA is irrelevant for robot

Bioinspir. Biomim. 10 (2015) 036003

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Figure 6. Comparison of inclination angle for the cases of static and non-static base. In case of static base, the DEA is aligned with the gravity (ϕ = 0 ). However, for non-static, accelerating base, the DEA (ϕ) varies and the VDI measures inclination angle (θ) from the DEA using equation (9).

Figure 7. Comparison of angular velocity ( β˙ ) and acceleration ( β¨ ) from VDI sensor (equation (8) and gyroscope) against the differentiated magnetic encoder readings.

Figure 8. (a) Model of a robot with point O modeled as a Euler 2–1 universal joint with (b) detail view of the mounted planar Vestibule Dynamic Inclinometer (pVDI) sensor. The hollow arrows indicate the sensitive direction of the dual-axis accelerometers ({L1, R1 }, {L 2 R 2 }). The tri-axial gyroscope is indicated by G.

7

Bioinspir. Biomim. 10 (2015) 036003

V Vikas and C Crane

equilibrium, thus, requiring only two independent parameters (equivalent universal-joint) to align the robot along the DEA. Hence, the point O is modeled as a universal joint capable of translating on the surface of contact as shown in figure 8(a). The body possesses two angular degrees-of-freedom, i.e., a minimum of two rotations are required to align the e 3 to E 3. Let R1 (θ), R 2 (ψ ) represent elementary rotation matrices [28, 31] that rotate a vector with angle θ , ψ about axis 1, 2 respectively. Let there be a vector (v ), that is measured in coordinate systems D (v D ) and B (v B ). The point O universal joint may involve 1–2 or 2–1 Euler angle rotation. For the remaining paper, universal joint with 2–1 Euler rotation is assumed as shown in figures 8(a). It is trivial to perform calculations for the other case given the solution procedure for the first case. So, for 2–1 Euler rotation from B to D D

B

v = R1 (θ) R 2 (ψ ) v .

(12)

Given v D, v B , the solution of θ , ψ can be written as [32] ⎛ ⎜ θ = cos−1 ⎜ ⎜ ⎝

v1B 2

2

( v1D ) + ( v3D )

(

⎞ ⎟ ⎟−λ ⎟ ⎠

)

where λ = atan2(v3D, v1D ), w = v1D sθ + v3D cθ . The angular velocity and acceleration represented B ( N ωeB , N B αe ) in terms of θ , ψ are ⎡ θ¨c − θ˙ s ψ˙ ⎤ ψ ⎢ ψ ⎥ N B ψ¨ αe = ⎢ ⎥ (15) ⎢ θ¨s + θ˙ c ψ˙ ⎥ ψ ⎦ ⎣ ψ

Here, it is desired to measure the orientation parameters—orientation Euler angles (θ , ψ ), angular velocity ( N ωB ) and angular acceleration ( N α B ). The planar Vestibule Dynamic Inclinometer (pVDI) extends the VDI for non-planar motion of the robot. It consists of four symmetrically placed dual-axis accelerometers and one triaxial gyroscope. Linear accelerometers {L1, R1 }, {L 2, R 2 } are placed symmetrically at distance of d1 2, d 2 2 along the e1, e 2 direction about point P and measure the acceleration along {e 3 , e1}, {e 3 , e 2} respectively as shown in figure 8(b) di di r O → Li = l e 3 − ei , r O → Ri = l e 3 + 2 2 ei i = 1, 2.

(16)

4.2. Orientation measurement—analytical solution The weighted difference and mean of the accelerometer readings and gyroscope reading can be written for i = 1, 2 N

ζi =

aRi − N aLi di

= Λei

(17)

8

aRi + N aLi

N

=

2l ζ4 =

aO + Λe 3 l

ωeB .

N

(18) (19)

The symmetry and kinematic relationships solves the angular acceleration as N

αeB = ⎡⎣ ζ23 − ζ42 · ζ43, − ζ13 + ζ41 · ζ43,

ζ12 − ζ21 ⎤T ⎥⎦ . 2

(20)

Now, it is possible to calculate g˜ e , i.e., g˜ expressed in B as

(

)

g˜ e = l ζ3 − ζ5

(21)

ζ5 = ⎡⎣ ζ41 · ζ43 + N α2B,

for

2 2 ⎤T ζ42 · ζ43 − N α1B, −ζ41 − ζ42 ⎦ . The unit vector ˆg˜ = g˜ g˜, where g˜ = ∣∣ g˜ ∣∣2 . As the aim is to align the e e e robot along the DEA, the unit vector of resultant acceleration of point of contact expressed in D can be written as g˜ˆ E = [0,0,1]T

g˜ˆ E = R1 (θ) R 2 (ψ ) g˜ˆ e .

(13)

ψ = atan2 −wv2B + v2D v3B, wv3B + v2D v2B , (14)

⎡ θ˙ c ⎤ ⎢ ψ⎥ N B ωe = ⎢ ψ˙ ⎥ , ⎢ θ˙ s ⎥ ⎣ ψ⎦

N

ζ3 =

(22)

Equation (22) is of the same form as equation (12). So, θ , ψ can be calculated as shown in equations (13) and (14). Using equations (15), (20) it is possible to calculate θ˙ , ψ˙ , θ¨, ψ¨ as ζ41 ζ43 or θ˙ = ζ42, ψ˙ = cψ sψ

(23)

ζ52 + ζ42 ζ43 ζ53 − ζ42 ζ41 or . θ¨ = ζ52, ψ¨ = cψ sψ

(24)

The approach assumes that the accelerometers are symmetrically placed on a beam. However, for lack of symmetry such that

( ) = 2 l ( e 3 + δi − ei ) for i = 1, 2.

r O → Ri − r O → Li = di 1 + δi + ei r O → Ri + r O → Li

(25)

Then, using equations (17)–(18) and (25), the corrected weighted difference and means (ζ1corr, ζ2corr, ζ3corr ) are

(

)

ζi corr = 1 + δi + ζi

for i = 1, 2

ζ3corr = ζ3 + δ1 −ζ1 + δ 2 −ζ2.

(26) (27)

4.3. Experiment and discussion The experimental setup illustrated in figure 9 attempts to emulate the rigid body modeled as an inverted pendulum shown in figure 8. The pVDI sensor is located as a distance l = 50 cm from the base joint. The universal joint at the base (equivalent point O) is fixed with two magnetic encoders (US Digital MA3A10–125-N) to measure the θ , ψ Euler angles. The pVDI inertial sensor comprises of four dual-axis linear MEMS accelerometers (ADXL335 with noise density 1 of 300 μg Hz− 2 rms) symmetrically placed at 11 mm

Bioinspir. Biomim. 10 (2015) 036003

V Vikas and C Crane

Figure 9. The pVDI experimental setup tries to emulate the modeled inverted pendulum in figure 8, such that l = 50 cm and d1 = d 2 = 22 cm . The universal joint has two magnetic encoders to measure the two Euler angles— θ , ψ .

about the center-line of the inverted pendulum i.e. d1 = d 2 = 22 cm . One triaxial MEMs gyroscope (two

filtering [33]. The experiment compares the inclination measurements obtained from pVDI sensor (equations (13), (14), (23), (24)) to the readings from the magnetic encoders. The plots of comparison of the inclination angle, angular velocity, and angular acceleration are shown in figures 10–12. The observations had standard errors of 0.033 deg , 0.036 deg , 0.187 deg s−1, 0.261 deg s−1, 1.264 deg s−2, −2 ¨ ˙ 1.049 deg s for θ , ψ , θ , ψ˙ , θ and ψ¨ respectively. Additionally, the standard deviations for the error distributions for Euler angles, angular velocity and angular accelerations (order θ , ψ ) were 0.376 deg , 0.285 deg , 5.536 deg s−1, 7.719 deg s−1, −2 −2 37.415 deg s and 31.039 deg s . The experimental results are good and validate the capability of pVDI to measure orientation parameters. They are very encouraging, however, lack of robustness in the system for measurement of angular acceleration was observed due to uncertainty in measurement of calibration parameters i.e. l , d1, d 2 and alignment of sensor coordinate systems to body coordinate system. This has been partially addressed in equations (26), (27). Nonetheless, calibration of distance parameters (l , d1, d 2) is viewed as the correct solution to this robustness problem and needs to be researched. Calculation of ζ5 (contributing to θ¨, ψ¨ ) has higher error as combines errors from both gyroscope (figure 11) and accelerometer measurements. This can be addressed by designing a gyroscope-free accelerometer based inclination sensor as calibration of gyroscopes is not as trivial as calibration of accelerometers.

1

LPY503AL with noise density of 0.014° s−1 Hz− 2 rms) was strapped onto the inverted pendulum to measure the angular velocity. The readings from the magnetic encoders are assumed to be ground truth for inclination angles (θ , ψ ) and differentiated to obtain the true Euler angular velocities (θ˙ , ψ˙ ) and accelerations (θ¨, ψ¨ ). The experiment was performed keeping the base fixed, i.e., N aO = g E1 (figure 8(a)). All the sensor measurements were sampled at 50 Hz and the noise was processed using low-pass forward–backward

5. Conclusion The presented research discusses dynamic equilibrium for rigid bodies modeled as inverted pendulums and design of two bio-inspired inertial sensors that measure orientation from dynamic equilibrium. The two symmetrically placed human ears provide design inspiration—‘symmetric’ placement of ‘multiple’ inertial sensors.

Figure 10. Plot of θ , ψ Euler angles obtained from pVDI sensors using equations (13), (14) against the readings from the magnetic encoders fixed inside the universal joint. The pVDI Euler angle measurements follow the encoder readings well with 0.035 deg , 0.331 deg mean standard error and deviation respectively.

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Figure 11. Plot of θ˙ , ψ˙ Euler angular velocities from differentiated magnetic encoder readings against those obtained from pVDI sensor using equation (23). The mean standard deviation and error are 0.224, 6.628 deg s−1

Figure 12. The comparison of θ¨, ψ¨ Euler angular accelerations obtained from pVDI (equation (24)) with double differentiated magnetic encoder readings yields mean standard deviation and error of 1.156, 34.227 deg s−2 , respectively.

The DEA is the axis along which the robot is at equilibrium. It is parallel to the direction of the resultant acceleration of the surface of contact (gravity, etc) and acts as the reference for measurement. The DEA ceases to exist when the resultant acceleration of the contact platform/surface is zero, i.e., zero gravity (e.g., free-fall). The DEA helps to explain the tendency of human body to lean backward, forward or sideways to align themselves to equilibrium position in acceleration varying environments. It also explains the surface-normal human posture when standing on a flat surface but not on an incline (gravity-aligned). The VDI and pVDI are human vestibular system motivated orientation measurement inertial sensors. They measure the orientation parameters—angular orientation (not relative change), angular velocity, angular acceleration and magnitude of acceleration of the surface of contact. The angular acceleration measurement is unique to these inertial sensors. The orientation measurements are independent of the acceleration of the surface of contact (gravity, etc), independent of body dynamics, not time-recursive, devoid of integration errors and valid for large inclination angles. This makes them useful in environments with varying gravity and accelerating platforms. The

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measured parameters and equations lay foundation to design of estimators (due to availability of angular acceleration, etc) and gyroscope-free inertial sensors. The sensor outputs are ideal control inputs for balancing of robots as the goal is to bring the robots to the equilibrium position (i.e., align it along the DEA). The experimental design of the VDI sensor consists of two symmetrically placed dual-axis MEMS linear accelerometers and one single axis MEMS gyroscope. While, for the pVDI sensor, the design consists of symmetrically placed four dual-axis MEMS linear accelerometers and one tri-axial MEMS gyroscope. The sensitive axes of the symmetrically placed dualaxis linear accelerometers for the pVDI is different from that of the analogous accelerometers in the case of the VDI. The experiments provide conceptual validation to the concept of the DEA (the non-static base) and the inclination parameters calculated using the analytical solution are analyzed.

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