Isotropic Optical Mouse Placement for Mobile Robot Velocity Estimation

Abstract

This paper presents the isotropic placement of multiple optical mice for the velocity estimation of a mobile robot. It is assumed that there can be positional restriction on the installation of optical mice at the bottom of a mobile robot. First, the velocity kinematics of a mobile robot with an array of optical mice is obtained and the resulting Jacobian matrix is analysed symbolically. Second, the isotropic, anisotropic and singular optical mouse placements are identified, along with the corresponding characteristic lengths. Third, the least squares mobile robot velocity estimation from the noisy optical mouse velocity measurements is discussed. Finally, simulation results for several different placements of three optical mice are given.

Keywords

Optical Mice Mobile Robot Velocity Estimation Isotropic Optical Mouse Placement Optimal Characteristic Length

1. Introduction

For the velocity estimation of a mobile robot, several attempts have been made to use optical mice that were originally invented as an advanced computer pointing device [1 –11]. In fact, an optical mouse is an inexpensive but high performance motion detection sensor with a sophisticated image processing engine inside. Optical mice installed at the bottom of a mobile robot, as shown in Figure 1, can detect the motions of a mobile robot traveling over a plane surface. Mobile robot velocity estimation using optical mice does not suffer from the problems of typical sensors: wheel slip in encoders, the line of sight in ultrasonic sensors and heavy computation in cameras.

Figure 1.

A prototype of a three optical mice array for mobile robot velocity estimation [4]

A pair of optical mice was proposed as a simple but viable means for carrying out mobile robot velocity estimation in the presence of wheel slip [1, 2]. Using redundant velocity measurements of two optical mice, a simple procedure for error detection and reduction in the mobile robot velocity estimation was developed [3]. The redundant number of optical mice was proposed to reduce the effect of the noisy velocity measurements of optical mice and to handle their partial malfunction [4]. Using the geometrical relationships between the optical mice, the calibration for systematic errors and the selection of reliable velocity measurements were carried out [5].

For a mobile robot with a circular base, a regular polygonal array of optical mice can be a natural and desirable choice for the placement of the optical mice. For instance, a pair of optical mice is placed so as to be symmetric around the centre of a mobile robot. Furthermore, N(≥ 3) optical mice are placed in a regular N -gonal array, the geometrical centre of which is coincident with the centre of a mobile robot [4]. However, there can be some restrictions on the installation of optical mice, if the mobile robot has a non-circular base or other structures are pre-installed on its base. With positional restriction on installation, a non-regular polygonal array of optical mice can be a better choice, compared with its regular polygonal counterpart [6].

The performance of an optical mouse array for mobile robot velocity estimation can be evaluated based on its Jacobian matrix. The Jacobian matrix maps the velocity of a mobile robot to the velocities of optical mice, which is a function of the installation positions of optical mice. Through the Jacobian matrix, the unit sphere in the optical mouse velocity space can be mapped into an ellipsoid in the mobile robot velocity space. For the optimal placement of optical mice, the volume of the ellipsoid can be one measure and its closeness to a sphere, the so-called isotropy, can be another measure. The concept of isotropy has been adopted for the optimal design of serial and parallel manipulators [7 –11], as well as, omnidirectional mobile robots [12, 13].

This paper presents the isotropic placement of optical mice for mobile robot velocity estimation. It is assumed that there can be positional restrictions on the installation of optical mice at the bottom of a mobile robot. This paper is organized as follows. Section 2 obtains the velocity kinematics of a mobile robot equipped with optical mice, and Section 3 analyses the resulting Jacobian matrix symbolically. Sections 4, 5, and 6 identify the isotropic, anisotropic, and singular optical mouse placements, along with the corresponding characteristic lengths. Section 7 discusses the least squares mobile robot velocity estimation from the noisy optical mouse velocity measurements. Section 8 gives the simulation results for several placements of three optical mice. Finally, the conclusions are presented in Section 9

2. Preliminary: Velocity Kinematics

The velocity of a mobile robot traveling on a plane can be estimated using the velocity measurements of N(≥ 2) optical mice installed at the bottom of a mobile robot. Figure 2 shows three coordinate frames that are used for the description of a mobile robot and the i^th optical mouse. Let O_W, X_W, and Y_W denote the origin, and the x and y axes of the world coordinate frame, respectively; let O_R, X_R, and Y_R denote the origin, and the x and y axes of the mobile robot coordinate frame, respectively; and, let O_i, X_i, and Y_i, i = 1, ···, N, denote the origin, the x and y axes of the i^th optical mouse coordinate frame, respectively. For a simple description, the following assumptions are made. 1) Two origins, O_W and O_R, are coincident with the centre, denoted by O, of a mobile robot. 2) The origin, O_i, i = 1, ···, N is coincident with the installation position, P_i, of the i^th optical mouse. 3) The mobile robot coordinate frame is aligned with the i^th optical mouse coordinate frame. The position vector, p_i = [x_i y_i]^t, i = 1, ···, N, of the i^th optical mouse can be expressed by

Figure 2.

Three coordinate frames for a mobile robot and the i^th optical mouse

p_{i} = [\begin{matrix} x_{i} \\ y_{i} \end{matrix}] = [\begin{matrix} ρ_{i} \times \cos (θ + φ_{i}) \\ ρ_{i} \times \sin (θ + φ_{i)} \end{matrix}], i = 1, \dots, N

(1)

In the above, θ is the angle of rotation of the mobile robot coordinate frame with reference to the world coordinate frame; ρ_i and ϕ_i are the polar coordinates of the installation position P_i of the i^th optical mouse. Note that the heading angle of a mobile robot is given by $(θ + \frac{π}{2})$ .

Let v_rx and v_ry be two linear velocity components of a mobile robot along the x axis and the y axis, respectively, and ω_r be its angular velocity component around the centre O of a mobile robot. Furthermore, let v_ix and v_iy, i = 1, ···, N be the lateral and longitudinal velocity measurements of the i^th optical mouse. The velocity relationship between a mobile robot and the i^th optical mouse can be presented by

v_{r x} - ω_{r} \times y_{i} = v_{i x}, i = 1, \dots, N

(2)

v_{r y} + ω_{r} \times x_{i} = v_{i y}, i = 1, \dots, N

(3)

From (2) and (3), the velocity kinematics of a mobile robot with an array of N optical mice can be obtained by [1, 4 –6, 14, 16, 17]

A v_{r} = v_{s}

(4)

In the above, v_r = [v_rx v_ry ω_r]^t ∊ R^3×1 represents the velocity vector of a mobile robot, v_s = [v₁^t v₂^t ··· v_N^t]^t ∊ R^2Nx1 represents the velocity vector of N optical mice, with v_i = [v_ix v_iy]^t, i = 1, ···, N being the velocity measurement of the i^th optical mouse; and, A representing the Jacobian matrix mapping v_r to v_S, given by [1, 4 –6, 14, 16, 17]

A = [\begin{matrix} \begin{matrix} \begin{matrix} 1 \\ 0 \\ 1 \end{matrix} \\ 0 \\ \begin{matrix} ⋮ \\ 1 \\ 0 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \\ 1 \\ \begin{matrix} ⋮ \\ 0 \\ 1 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} - y_{1} \\ x_{1} \\ {- y}_{2} \end{matrix} \\ x_{2} \\ \begin{matrix} ⋮ \\ {- y}_{N} \\ x_{N} \end{matrix} \end{matrix} \end{matrix}] ∊ R^{2 N \times 3}

(5)

Note that the expression of A is very simple as a function of the positions of N optical mice, p_i = [x_i y_i]^t, i = 1, ···, N.

3. Symbolic Analysis of the Jacobian Matrix

In the velocity kinematics of (4), the velocity vector of a mobile robot, v_r, is composed of two linear and one angular component, while the velocity vector of N optical mice, v_s, is composed of a total of 2N linear components. To eliminate the physical inconsistency among velocity components, the characteristic length, denoted by L, can be introduced [8]:

\hat{A} {\hat{v}}_{r} = v_{s}

(6)

where ${\hat{v}}_{r} = {[v_{r x} v_{r y} (L \times ω_{r})]}^{t} ∊ R^{3 \times 1}$ and

\hat{A} = [\begin{matrix} \begin{matrix} 1 & 0 & - \frac{1}{L} \times y_{1} \end{matrix} \\ \begin{matrix} \begin{matrix} 0 & 1 & \frac{1}{L} \times x_{1} \end{matrix} \\ \begin{matrix} \begin{matrix} 1 & 0 & - \frac{1}{L} \times y_{2} \end{matrix} \\ \begin{matrix} \begin{matrix} 0 & 1 & \frac{1}{L} \times x_{2} \end{matrix} \\ \begin{matrix} \begin{matrix} ⋮ & ⋮ & ⋮ \end{matrix} \\ \begin{matrix} \begin{matrix} 1 & 0 & - \frac{1}{L} \times y_{N} \end{matrix} \\ \begin{matrix} 0 & 1 & \frac{1}{L} \times x_{N} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}] ∊ R^{2 N \times 3}

(7)

Note that all elements of $\hat{A}$ are physically dimensionless. From (7), ${\hat{A}}^{t} \hat{A}$ can be written as

{\hat{A}}^{t} \hat{A} = N \times [\begin{matrix} 1 & 0 & - \frac{C_{y}}{L} \\ 0 & 1 & \frac{C_{x}}{L} \\ - \frac{C_{y}}{L} & \frac{C_{x}}{L} & \frac{R^{2}}{L^{2}} \end{matrix}] ∊ R^{3 \times 3}

(8)

where

C_{x} = \frac{1}{N} \sum_{i = 1}^{N} x_{i}

(9)

C_{y} = \frac{1}{N} \sum_{i = 1}^{N} y_{i}

(10)

R = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} (x_{i}^{2} + y_{i}^{2})} = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} ρ_{i}^{2}}

(11)

In the above, C_x and C_y, respectively, represent the averages of the x and y coordinates of the position vectors, p_i = [x_i y_i]^t, i = 1, ···, N, of N optical mice, and R represents the root mean square of the distances of N optical mice from the centre O of a mobile robot. Using (8), the characteristic polynomial of A^tA is given by

det (λ I_{3} - A^{t} A) = 0

(12)

\begin{matrix} (λ - N) \{λ^{2} - N (1 + \frac{R^{2}}{L^{2}}) λ \\ + N^{2} (\frac{R^{2}}{L^{2}} - \frac{C_{x}^{2} + C_{y}^{2}}{L^{2}})\} = 0 \end{matrix}

(13)

where I₃ represents the 3 × 3 identity matrix. From (13), three eigenvalues of A^tA, denoted by λ₁, λ₂, and λ₃, are obtained by [6]

\begin{matrix} λ_{1} = \frac{N}{2} \{(1 + \frac{R^{2}}{L^{2}}) \\ + \sqrt{{(1 - \frac{R^{2}}{L^{2}})}^{2} + \frac{4 (C_{x}^{2} + C_{y}^{2})}{L^{2}}}\} \end{matrix}

(14)

λ_{2} = N

(15)

\begin{matrix} λ_{3} = \frac{N}{2} \{(1 + \frac{R^{2}}{L^{2}}) \\ - \sqrt{{(1 - \frac{R^{2}}{L^{2}})}^{2} + \frac{4 (C_{x}^{2} + C_{y}^{2})}{L^{2}}}\} \end{matrix}

(16)

The following inequality relationships exist between the three eigenvalues, λ₁, λ₂, and λ₃ [6]:

λ_{3} \leq λ_{2} (= N) \leq λ_{1}

(17)

In the case of $\frac{R^{2}}{L^{2}} \leq 1$ , we have

\begin{matrix} (1 + \frac{R^{2}}{L^{2}}) + \sqrt{{(1 - \frac{R^{2}}{L^{2}})}^{2} + \frac{4 (C_{x}^{2} + C_{y}^{2})}{L^{2}}} \\ \geq (1 + \frac{R^{2}}{L^{2}}) + (1 - \frac{R^{2}}{L^{2}}) = 2 \end{matrix}

(18)

And, in the case of $\frac{R^{2}}{L^{2}} > 1$ , we have

\begin{matrix} (1 + \frac{R^{2}}{L^{2}}) + \sqrt{{(1 - \frac{R^{2}}{L^{2}})}^{2} + \frac{4 (C_{x}^{2} + C_{y}^{2})}{L^{2}}} \\ \geq (1 + \frac{R^{2}}{L^{2}}) + (\frac{R^{2}}{L^{2}} - 1) = 2 \times \frac{R^{2}}{L^{2}} > 2 \end{matrix}

(19)

Using (18) and (19), from (14) and (15), it follows that λ₁ ≥ λ₂ (= N), regardless of the values of C_x, C_y, and R², as well as L. Similar to the above, it can be also shown that λ₃ ≤ λ₂(= N). Note that λ₁ and λ₃ are the largest and smallest eigenvalues of A^tA, while λ₂ is its middle eigenvalue which remains constant as N.

4. Isotropic Optical Mouse Placement

For the 2N × 3 Jacobian matrix $\hat{A}$ with N ≥ 2, the condition number can be defined by [19]

κ = \frac{σ_{1}}{σ_{3}} = \sqrt{\frac{λ_{1}}{λ_{3}}}

(20)

where σ₁ and σ₃ represent, respectively, the largest and smallest singular values of $\hat{A}$ , λ₁ and λ₃, represent, respectively, the largest and smallest eigenvalues of ${\hat{A}}^{t} \hat{A}$ . Note that the condition number κ can have values from unity to infinity. The Jacobian matrix $\hat{A}$ is isotropic when κ = 1, and $\hat{A}$ is singular when κ = ∞.

The placement of N optical mice is said to be isotropic, if the isotropy of the 2N × 3 Jacobian matrix $\hat{A}$ can be achieved:

{\hat{A}}^{t} \hat{A} = N \times I_{3}

(21)

for which ${\hat{A}}^{t} \hat{A}$ has three identical eigenvalues of magnitude N, λ₁ = λ₂ = λ₃ = N, so that the condition number becomes unity, κ = 1. From (8) and (21), the isotropy conditions for $\hat{A}$ are given by

C_{x} = \frac{1}{N} \sum_{i = 1}^{N} x_{i} = 0

(22)

C_{y} = \frac{1}{N} \sum_{i = 1}^{N} y_{i} = 0

(23)

\frac{R^{2}}{L^{2}} = \frac{1}{L^{2}} \times \frac{1}{N} \sum_{i = 1}^{N} (x_{i}^{2} + y_{i}^{2}) = 1

(24)

In the above, (22) and (23) indicate that the geometrical centre of N optical mice coincides with the centre O of a mobile robot. Furthermore, (24) indicates that the squared value of the characteristic length should be equal to the average of the squared distances of N optical mice from the centre O of a mobile robot.

Using (1), (22) and (23) can be written as:

\sum_{i = 1}^{N} p_{i} = 0

(25)

Let S_N be the isotropic set of the position vectors of N optical mice, satisfying (25):

S_{N} = \{p_{i}, i = 1, \dots, N |\sum_{i = 1}^{N} p_{i} = 0\}

(26)

For given N(≥ 2) optical mice, Figure 3 shows the isotropic sets of N position vectors, p_i, i = 1, ···, N. In the case of N = 2 shown in Figure 3(a), the isotropic set S₂ can be parameterized by two variables, (y₁, θ); in the case of N = 3 shown in Figure 3(b), the isotropic set S₃ can be parameterized by four variables, (y₁, x₂, y₂, θ); and, in the case of N(≥ 4) shown in Figures 3(c) and 3(d), the isotropic set S_N can be parameterized by 2(N − 1) variables, (y₁, x₂, y₂, ···, x_N-1, y_N-1, θ). Note that the rotation of the isotropic set of N position vectors by the angle θ with respect to the centre O of a mobile robot is also isotropic:

Figure 3.

The isotropic set of the position vectors of N optical mice: (a) N = 2, (b) N = 3, (c) N = 4, and (d) N ≥ 4.

\sum_{i = 1}^{N} R p_{i} = R (\sum_{i = 1}^{N} p_{i}) = 0

(27)

where

R = [\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}]

(28)

represents the rotation matrix.

Since the union of two isotropic sets is also isotropic, new isotropic sets for N(≥ 4) position vectors can be obtained from existing isotropic sets known already:

S_{N} = S_{N_{1}} ⋃ S_{N_{2}}

(29)

where N = N₁ + N₂, 2 ≤ N₁, N₂ ≤ N − 2. For N = 5 optical mice, Figure 4(a) shows the isotropic set S₅, which is obtained as the union of S₂ and S₃. However, note that (29) cannot produce all possible isotropic sets of N position vectors, since $\sum_{j = 1}^{N_{1}} p_{j} = \sum_{j = N_{1} + 1}^{N} p_{j} = 0$ is sufficient but not necessary for $\sum_{i = 1}^{N} p_{i} = 0$ . It should be mentioned that the simplest isotropic set of N position vectors is a regular polygon, for which

ρ_{1} = ρ_{2} = \dots = ρ_{N}

(30)

φ_{2} - φ_{1} = φ_{3} - φ_{2} = \dots = φ_{1} - φ_{N} = \frac{360 °}{N}

(31)

Figure 4.

Two isotropic sets of N = 5 position vectors: (a) the union of S₂ and S₃, and (b) a regular pentagon

Figure 4(b) shows the isotropic placement of N=5 position vectors, which is a regular pentagon.

Once the isotropic placement of N optical mice, denoted by p_i^* = [x_i^* y_i^*]^t, i = 1, ···, N, is determined from (22) and (23), the value of the characteristic length L, required for the isotropy of the Jacobian matrix $\hat{A}$ , can be determined, from (24):

L^{*} = R^{*} = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} (x_{i}^{* 2} + y_{i}^{* 2})}

(32)

which is called the optimal characteristic length. Note that the optimal characteristic length L^* is the root mean square of the distances of N optical mice from the centre O of a mobile robot.

5. Anisotropic Optical Mouse Placement

For a given placement of N optical mice, it may be impossible to achieve the isotropy of the 2N × 3 Jacobian matrix $\hat{A}$ . Seen from (16), the smallest eigenvalue of A^tA, λ₃, can be zero, and thus we consider the condition index, defined by

γ = \frac{λ_{3}}{λ_{1}}

(33)

which is the inverse of the condition number κ of A^tA, given by (20). Note that the condition index γ can have values between zero and unity, where ${\hat{A}}^{t} \hat{A}$ is isotropic when γ = 1, and ${\hat{A}}^{t} \hat{A}$ is singular when γ = 0. For the velocity kinematics, given by (8), the condition index γ, given by (33), can be used a measure of the relative round-off error amplification of ${\hat{v}}_{r}$ with respect to the relative round-off error of v_s [8, 19]. To reduce the effect of relative round-off error amplification, the value of γ needs to be maximized, that is, the system under consideration should be made close to isotropy.

For the placement of N optical mice, it is desirable to make the value of γ as large as possible. Using (14) and (16), (33) can be written as

γ = \frac{A - \sqrt{B}}{A + \sqrt{B}}

(34)

with

A = 1 + \frac{R^{2}}{L^{2}}

(35)

B = {(1 - \frac{R^{2}}{L^{2}})}^{2} + \frac{4 (C_{x}^{2} + C_{y}^{2})}{L^{2}}

(36)

Differentiating the condition index γ, given by (34), with respect to L, we have

\frac{\partial γ}{\partial L} = \frac{(A + \sqrt{B}) (A^{'} - \frac{B^{'}}{2} \sqrt{B}) - (A - \sqrt{B}) (A^{'} + \frac{B^{'}}{2} \sqrt{B})}{{(A + \sqrt{B})}^{2}}

(37)

where

A^{'} = \frac{\partial A}{\partial L} = 2 L

(38)

B^{'} = \frac{\partial B}{\partial L} = 4 (L^{2} - R^{2}) \times L + 8 (C_{x}^{2} + C_{y}^{2}) \times L

(39)

Setting $\frac{\partial γ}{\partial L}$ equal to zero, we have

2 A^{'} \times B = A \times B^{'}

(40)

Plugging (35), (36), (38), and (39) into (40), it follows that

(L^{2} - R^{2}) \times \{R^{2} - (C_{x}^{2} + C_{y}^{2})\} = 0

(41)

As will be shown later,

R^{2} \neq C_{x}^{2} + C_{y}^{2}

(42)

unless ${\hat{A}}^{t} \hat{A}$ is singular. From (41) and (42), using (11), we obtain

L^{2} - R^{2} = 0

(43)

which results in

L^{#} = R^{#} = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} (x_{i}^{2} + y_{i}^{2})}

(44)

which is called the suboptimal characteristic length. It should be noted that the expression of the suboptimal characteristic length L^#, given by (44), is the same as that of the optimal characteristic length L^*, given by (32).

With the suboptimal characteristic length L^# known, the maximum value of the condition index γ that can be achieved for a given anisotropic optical mouse placement can be obtained. Plugging (43) and (44) into (35) and (36) and using (34), we obtain

γ^{#} = \frac{L^{#} - \sqrt{C_{x}^{2} + C_{y}^{2}}}{L^{#} + \sqrt{C_{x}^{2} + C_{y}^{2}}}

(45)

which is called the maximal condition index. Note that the maximal condition index γ^# can have values between zero and unity, where γ^# = 1 when C_x = C_y = 0, and γ^# = 0 when $L^{#} = \sqrt{C_{x}^{2} + C_{y}^{2}} (= R^{#})$ .

6. Singular Optical Mouse Placement

The placement of N optical mice is said to be singular, if ${\hat{A}}^{t} \hat{A}$ falls into singularity, that is, the smallest eigenvalue of ${\hat{A}}^{t} \hat{A}$ , λ₃, becomes zero:

λ_{3} = 0

(46)

for which the condition index becomes zero, γ = 0. From (16) and (46), we have

(1 + \frac{R^{2}}{L^{2}}) = \sqrt{{(1 - \frac{R^{2}}{L^{2}})}^{2} + \frac{4 (C_{x}^{2} + C_{y}^{2})}{L^{2}}}

(47)

which leads to

R^{2} = C_{x}^{2} + C_{y}^{2}

(48)

Plugging (9)–(11) into (48), we have

\begin{matrix} (N - 1) \sum_{i = 1}^{N} (x_{i}^{2} + y_{i}^{2}) \\ = 2 \sum_{j = 1}^{N - 1} \sum_{k = j + 1}^{N} (x_{j} x_{k} + y_{j} y_{k}) \end{matrix}

(49)

Using (1), (49) can be written as

\begin{matrix} (N - 1) \sum_{i = 1}^{N} ρ_{i}^{2} \\ - 2 \sum_{j = 1}^{N - 1} \sum_{k = j + 1}^{N} ρ_{j} {\times ρ}_{k} \times \cos ψ_{j, k} = 0 \end{matrix}

(50)

where

\begin{matrix} ψ_{j, k} = φ_{k} - φ_{j}, j = 1, \dots, N - 1, \\ k = (j + 1), \dots, N \end{matrix}

(51)

represents the angle between two position vectors, p_j and p_k, j = 1, ···, N − 1, k = (i + 1), ···, N. Now, (50) can be rearranged into

\sum_{j = 1}^{N - 1} \sum_{k = j + 1}^{N} (ρ_{j}^{2} + ρ_{k}^{2} - 2 ρ_{j} \times ρ_{k} \times \cos ψ_{j, k}) = 0

(52)

where $(N - 1) \sum_{i = 1}^{N} ρ_{i}^{2} = \sum_{j = 1}^{N - 1} \sum_{k = j + 1}^{N} (ρ_{j}^{2} + ρ_{k}^{2})$ is used.

Since

\begin{array}{l} 0 & \leq & {(ρ_{j} - ρ_{k})}^{2} \\ \leq & (ρ_{j}^{2} + ρ_{k}^{2} - 2 ρ_{j} \times ρ_{k} \times cos ψ_{j, k}) \\ \leq & {(ρ_{j} + p_{k})}^{2}, j = 1, \dots, N - 1, k = (i + 1), \dots, N \end{array}

(53)

we have

\begin{array}{l} 0 \leq \sum_{j = 1}^{N - 1} \sum_{k = j + 1}^{N} {(ρ_{j} - ρ_{k})}^{2} \\ \leq \sum_{j = 1}^{N - 1} \sum_{k = j + 1}^{N} (ρ_{j}^{2} + ρ_{k}^{2} - 2 ρ_{j} \times ρ_{k} \times \cos ψ_{j, k}) \\ \leq \sum_{j = 1}^{N - 1} \sum_{k = j + 1}^{N} {(ρ_{j} + ρ_{k})}^{2} \end{array}

(54)

In the above, the equality holds, when $\cos ψ_{j, k} = 1$ or $\cos ψ_{j, k} = - 1$ , j = 1, ···, N − 1, k = (i + 1), ···, N. In the former case, for which

\begin{matrix} ψ_{j, k} = 0, j = 1, \dots, N - 1, \\ k = (i + 1), \dots, N \end{matrix}

(55)

and, from (52), we have

\sum_{j = 1}^{N - 1} \sum_{k = j + 1}^{N} {(ρ_{j} - ρ_{k})}^{2} = 0

(56)

which results in

ρ_{1} = ρ_{2} = \dots = ρ_{N}

(57)

Note that (55) and (57) indicate that N optical mice are placed at the same position on a mobile robot. Next, in the latter case, for which

\begin{matrix} ψ_{j, k} = 180^{°}, j = 1, \dots, N - 1, \\ k = (i + 1), \dots, N \end{matrix}

(58)

and, from (52), we have

\sum_{j = 1}^{N - 1} \sum_{k = j + 1}^{N} {(ρ_{j} + ρ_{k})}^{2} = 0

(59)

which results in

ρ_{1} = ρ_{2} = \dots = ρ_{N} = 0

(60)

(60) indicates that N optical mice are placed at the centre O of a mobile robot. At both singular placements of N optical mice, one given by (55) & (57) and the other given by (60), it should be noted that the rank of the Jacobian matrix $\hat{A}$ drops to two.

7. Least Squares Velocity Estimation

Based on the velocity kinematics of (6), the mobile robot velocity can be estimated from the noisy velocity measurements v_s of N optical mice by

{\tilde{\hat{v}}}_{r} = [\begin{matrix} {\tilde{v}}_{r x} \\ {\tilde{v}}_{r y} \\ L \times {\tilde{ω}}_{r} \end{matrix}] = {\hat{A}}^{+} v_{s}

(61)

where

{\hat{A}}^{+} = {({\hat{A}}^{t} \hat{A})}^{- 1} {\hat{A}}^{t} ∊ R^{3 \times 2 N}

(62)

Note that (61) with (62) represents the least squares solution to (6), which minimizes $‖\hat{A} {\hat{v}}_{r} - v_{s}‖$ [19].

Let us consider the least square mobile robot velocity estimation, for the isotropic placement of N optical mice, with the isotropic position vectors p_i^* = [x_i^* y_i^*]^t, i = 1, ···, N, and the optimal characteristic length L^*. Plugging (7), (21), and (32) into (62), we have

\begin{matrix} {\hat{A}}^{+} \\ = \frac{1}{N} [\begin{matrix} \begin{matrix} 1 \\ 0 \\ - \frac{y_{1}}{R^{*}} \end{matrix} & \begin{matrix} \begin{matrix} 0 \\ 1 \\ \frac{x_{1}}{R^{*}} \end{matrix} & \begin{matrix} \begin{matrix} 1 \\ 0 \\ - \frac{y_{2}}{R^{*}} \end{matrix} & \begin{matrix} \begin{matrix} 0 \\ 1 \\ \frac{x_{2}}{R^{*}} \end{matrix} & \begin{matrix} \begin{matrix} \dots \\ \dots \\ \dots \end{matrix} & \begin{matrix} \begin{matrix} 1 \\ 0 \\ - \frac{y_{N}}{R^{*}} \end{matrix} & \begin{matrix} 0 \\ 1 \\ \frac{x_{N}}{R^{*}} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}] \end{matrix}

(63)

Using (63) from (61), three estimated velocity components of a mobile robot are obtained by

{\tilde{v}}_{r x} = \frac{1}{N} \sum_{i = 1}^{N} v_{i x}

(64)

{\tilde{v}}_{r y} = \frac{1}{N} \sum_{i = 1}^{N} v_{i y}

(65)

{\tilde{ω}}_{r} = \frac{1}{N} \sum_{i = 1}^{N} ω_{i}

(66)

where

ω_{i} = \frac{1}{R^{* 2}} \times (- y_{i}^{*} \times v_{i x} + x_{i}^{*} \times v_{i y}), i = 1, \dots, N

(67)

represents the angular velocity component experienced by the i^th optical mouse, which is equivalent to the velocity measurement v_i = [v_ix v_iy]^t, i = 1, ···, N. Seen from (64)–(66), two linear and one angular velocity component of a mobile robot can obtained by the averages of the corresponding components of N optical mice. Note that such computational simplicity in least squares mobile robot velocity estimation is attributed to the isotropic placement of N optical mice.

Now, let us discuss the role of the characteristic length L in least squares mobile robot velocity estimation, given by (61) with (62), which involves the inversion of ${\hat{A}}^{t} \hat{A}$ . Seen from (8), it is apparent that the magnitude of the characteristic length L will affect the condition index γ of ${\hat{A}}^{t} \hat{A}$ , given by (33), for a given optical mouse placement, p_i = [x_i y_i]^t, i = 1, ···, N. For instance, if L is set to be too small, ${\hat{A}}^{t} \hat{A}$ comes close to singularity, that is, γ → 0. This may lead to numerical instability during the inversion process of ${\hat{A}}^{t} \hat{A}$ , so the accuracy of the estimated mobile robot velocity may be unacceptably poor. On the other hand, the proper value of L, most preferably L = L^#, given by (44), can improve the conditioning of ${\hat{A}}^{t} \hat{A}$ , even when a given optical mouse placement is near-singular.

8. Simulation Results

Suppose that three optical mice (N = 3) are placed on an elliptical path, given by

Ω = \{(x, y) | {(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} = 1\}

(68)

where the first optical mouse is fixed on the principal axis along the y axis, but the second and third optical mice that are symmetric with respect to the y axis, are movable, as shown in Figure 5:

\begin{matrix} p_{1} = [\begin{matrix} x_{1} \\ y_{1} \end{matrix}] = [\begin{matrix} 0 \\ b \end{matrix}], p_{2} = [\begin{matrix} x_{2} \\ y_{2} \end{matrix}] = [\begin{matrix} - a \times \cos α \\ b \times \sin α \end{matrix}], \\ p_{3} = [\begin{matrix} x_{3} \\ y_{3} \end{matrix}] = [\begin{matrix} a \times \cos α \\ b \times \sin α \end{matrix}] \end{matrix}

(69)

Figure 5.

The symmetrical placement of three optical mice along an elliptical path

with a = 25 cm, b = 16 cm, and 0° ≤ α ≤ 90°. Using (22) and (23), the isotropic optical mouse placement can be found at α^* = 120°, corresponding to ((ϕ₁^*, ϕ₂^*, ϕ₃^*) = (90°, 210°, 330°). Note that three optical mice form an equilateral triangle in general, but they will form a regular triangle if the elliptical path Ω becomes circular, that is, a = b. Furthermore, using (32), the optimal characteristic length is obtained by $L^{*} = \sqrt{\frac{3}{2 N} (a^{2} + b^{2})} = 20.99 c m$ . On the other hand, using (55) and (57), the singular optical mouse placement can be found as (ϕ₁, ϕ₂, ϕ₃) = (90°, 90°, 90°). For a given optical mouse placement, let us examine the dependency of the condition index y, given by (33), on the characteristic length L. For comparison, four optical mouse placements are chosen: one isotropic placement, (ϕ₁^*, ϕ₂^*, ϕ₃^*) = (90°, 210°, 330°), denoted by Π₁, two anisotropic placements, (ϕ₁^*, ϕ₂^*, ϕ₃^*) = (90°, 180°, 360°). and (90°, 270°, 270°), denoted by Π₂ and Π₃, and one near-singular placement, (ϕ₁, ϕ₂, ϕ₃) = (90°, 100°, 80°), denoted by Π₄. As the characteristic length L increases from 1 to 30cm, Figure 6 shows the changes of the condition index γ, according to the value of the characteristic length L, for four placements, Π₁, Π₂, Π₃, and Π₄. Table 1 lists four suboptimal characteristic lengths L^#'s, and the corresponding maximal condition indices γ^#'s. Note that the results listed in Table 1 are the same as those computed using (44) and (45). In the case of the isotropic placement Π₁, γ^# becomes unity at L^# = L^* = 20.99 cm; in the cases of the anisotropic placements, Π₂ and Π₃, γ^# is about 0.5∼0.6; and, in the case of the near-singular placement Π₄, γ^# becomes very small. As the angle α increases from 0° to 180°, Figure 7 shows the changes of the suboptimal characteristic length L^# and the corresponding maximal condition index γ^#. Starting from L^# = b = 16 cm, the suboptimal characteristic length reaches a maximum of $L^{#} = \sqrt{\frac{1}{3} (2 a^{2} + b^{2})} = 22.41 c m$ , and then it decreases, becoming L^# = 16 cm. Note that the plot of L^# is symmetric with respect to the γ axis. Starting from γ^# ≃ 0cm, the maximal condition index reaches its maximum of γ^# = 1 cm, and then it decreases, becoming γ^# = 0.5 cm.

Figure 6.

shows the changes of the condition index γ, according to the value of the characteristic length L, for four optical mouse placements, Π₁, Π₂, Π₃, and Π₄.

Table 1.

The suboptimal characteristic length L^# and the corresponding maximal condition index γ^#

	Π₁	Π₂	Π₃	Π₄
L^# [cm]	20.99	22.41	16	16.23
γ^#	1	0.6155	0.5	0.0122

Figure 7.

The changes of the suboptimal characteristic length L^# and the corresponding maximal condition index γ^#, according to the value of the angle α.

Next, let us examine the least squares velocity estimation of a mobile robot for a given placement of three optical mice. Let us assume that a mobile robot is commanded to move forward along the γ axis at a velocity of v_rx = 0 cm/sec, v_ry = 20 cm/sec, and ω_r = 0 deg/sec. To simulate the noisy velocity measurements of three optical mice, normally distributed random numbers, n_ix and n_iy, i = 1,2,3, with mean 0 cm/sec and variance 0.2(cm/sec)² are added, independently, to the nominal values of the x and y velocity components of each optical mouse. With v_r = [0 20 0]^t, using (4), the noisy velocity measurements of three optical mice are obtained by

v_{s} = A v_{r} + n_{s}

(70)

where n_s = [n₁^t n₂^t n₃^t]t ∊ R^6×1 with n_i = [n_ix n_iy]^t, i = 1,2,3, represents the random noise vector experienced by three optical mice. For the isotropic optical mouse placement Π₁, Figure 8 shows the velocity measurements of the first optical mouse, and the resulting least squares velocity estimation of a mobile robot, using (61). Note that a total of 10,000 samples are taken in our simulation, but for better visibility, only 500 samples are plotted in Figure 8. Overall, it can be observed that the effects of noisy velocity measurements of three optical mice are reduced significantly. The noise levels of two linear components, ${\tilde{v}}_{r x}$ and ${\tilde{v}}_{r y}$ , of the estimated mobile robot velocity amount to about 58% of those of two linear velocity components of each optimal mouse.

Figure 8.

In the case of the isotropic optical mouse placement Π₁, the least squares mobile robot velocity estimation for a linear velocity of a mobile robot: (a) the measured velocity components, v_1x and v_1y, and (b) the estimated velocity components, ${\tilde{v}}_{r x}$ , ${\tilde{v}}_{r y}$ , $L^{*} \times {\tilde{ω}}_{r}$ , and ${\tilde{ω}}_{r}$ .

For four optical mouse placements, Π₁, Π₂, Π₃, and Π₄, Table 2 lists the root mean square errors of four components of the estimated mobile robot velocity, denoted by $R M S E ({\tilde{v}}_{r x})$ , $R M S E ({\tilde{v}}_{r y})$ , $R M S E (L^{#} \times {\tilde{ω}}_{r})$ , and RMSE(ω_r). For fair comparisons between the four different placements, the same random noise vector n_s is used repeatedly to generate their noisy velocity measurements using (70). From Table 2, the following observations can be made. In the cases of ${\tilde{v}}_{r x}$ and ${\tilde{ω}}_{r}$ , the root mean square errors tend to increase, as the optical mouse placement moves away from the isotropic one to the singular one. On the other hand, in the case of ${\tilde{v}}_{r y}$ , the root mean square error remains the same for four optical mouse placements. This results from the fact that ${\hat{A}}^{t} \hat{A}$ has the middle eigenvalue λ₂ = N(= 3) with the corresponding eigenvector given by u₂ = [0 1 0]^t, for all optical mouse placements. As the angle α increases from 0° to 180°, Figure 9 shows the changes of the root mean square errors of ${\tilde{v}}_{r x}, {\tilde{v}}_{r y}, L \times {\tilde{ω}}_{r}$ and ${\tilde{ω}}_{r}$ . From Table 2 and Figure 9, the following observations can be made. 1) In the case of ${\tilde{v}}_{r x}$ , the root mean square error decreases rapidly for 1° ≤ α ≤ 50°, but its change becomes slow for 50° ≤ α ≤ 180° with its minimum of 0.1151 cm/sec occurring at α = 122°. A similar pattern of change can be observed for $L^{#} \times {\tilde{ω}}_{r}$ , with its minimum of 0.8638 cm/sec occurring at α = 120°. 2) In the case of ${\tilde{ω}}_{r}$ , the changing pattern is similar to v_rx, but its minimum of 2.2652 deg/sec occurs at α = 99° not at α = 120°. This is because the isotropy of ${\hat{A}}^{t} \hat{A}$ is directly related to the least squares solution ${\tilde{\hat{v}}}_{r} = {[{\tilde{v}}_{r x} {\tilde{v}}_{r y} L \times {\tilde{ω}}_{r}]}^{t}$ , but not to ${\tilde{v}}_{r} = {[{\tilde{v}}_{r x} {\tilde{v}}_{r y} {\tilde{ω}}_{r}]}^{t}$ . 3) In the case of v_ry, the root mean square error remains the same regardless of the value of α. This is because ${\hat{A}}^{t} \hat{A}$ has a constant eigenvalue of λ₂ = 3 along the direction of u₂ = [0 1 0]^t, independently of α.

Table 2.

The root mean square errors of the measured and estimated velocity components

RMSE(·)	Π₁	Π₂	Π₃	Π₄
v_1x [cm/sec]	0.1994	0.1994	0.1994	0.1994
v_1y [cm/sec]	0.1992	0.1992	0.1992	0.1992
${\tilde{v}}_{r x} [cm/sec]$	0.1151	0.1188	0.1218	05278
${\tilde{v}}_{r y} [cm/sec]$	0.1152	0.1152	0.1152	0.1152
$L^{#} \times {\tilde{ω}}_{r} [c m / s e c]$	0.8683	03944	0.9304	3.9864
${\tilde{ω}}_{r} [d e g / s e c]$	23704	22872	33318	140728

Figure 9.

The changes of the root mean square errors of v_rx, v_ry, L^# × ω_r, and ω_r, according to the value of the angle α.

Now, assume that a mobile robot is commanded to rotate about the z axis at a velocity of v_rx = v_ry = 0 cm/sec and ω_r = 1 rad/sec (= 57.3 deg/sec). The same random noise vector n_s is used for both cases of the linear velocity and angular velocity of a mobile robot, for fair comparisons. In the case of angular velocity, the heading angle of a mobile robot, (θ + Π/2), keeps on changing, so that the position vectors of the optical mice change continuously, which results in changes to the Jacobian matrix $\hat{A}$ , seen from (1) and (5). Note that in the case of linear velocity, (θ + Π/2) is kept constant and $\hat{A}$ remains the same.

For the isotropic optical mouse placement Π₁, Figure 10 shows the velocity measurements of the first optical mouse, and the resulting least squares velocity estimation of a mobile robot. Seen from Figure 10(a), the plots of v_1x and v_1y are noisy sinusoids, with the same amplitude of ρ_i = b(= 16cm) but there is a phase difference of 90° between them; the plots of v_2x and v_2y are also noisy sinusoids, with the same amplitude of $ρ_{i} = b (= 16 c m)$ but there is also a phase difference of 90°. From Figure 10(b), it can be observed that the noise levels of ${\tilde{v}}_{r x}$ , ${\tilde{v}}_{r y}$ and $R \times {\tilde{ω}}_{r}$ are reduced to about 58% of those of v_ix and v_iy, i = 1,2.

Figure 10.

In the case of the isotropic optical mouse placement Π₁, the least squares estimation for an angular velocity of a mobile robot: (a) the measured velocity components, v_1x, v_1y, v_2x and v_2y, and (b) the estimated velocity components, ${\tilde{v}}_{r x}$ , ${\tilde{v}}_{r y}$ , $L^{*} \times {\tilde{ω}}_{r}$ , and ${\tilde{ω}}_{r}$

For four optical mouse placements, Π₁, Π₂, Π₃, and Π₄, Table 3 lists the root mean square errors of four components of the estimated mobile robot velocity, denoted by $R M S E ({\tilde{v}}_{r x})$ , $R M S E ({\tilde{v}}_{r y})$ , $R M S E (L^{#} \times {\tilde{ω}}_{r})$ and $R M S E ({\tilde{ω}}_{r})$ . From Table 3, it can be observed that in the case of ${\tilde{v}}_{r y}$ , as well as in the cases of ${\tilde{v}}_{r x}$ and ${\tilde{ω}}_{r}$ , the root mean square errors tend to increase, as the optical mouse placement moves away from the isotropic placement to the singular one. This is in contrast to the observation made from Table 2 that the root mean square error of ${\tilde{v}}_{r y}$ remains the same regardless of the optical mouse placements. As mentioned above, in the case of the angular velocity of a mobile robot, all three eigen vectors of ${\hat{A}}^{t} \hat{A}$ , u_i, i = 1,2,3, are subject to change, which results in similar patterns of changes in all three cases, ${\tilde{v}}_{r x}$ , ${\tilde{v}}_{r y}$ and ${\tilde{ω}}_{r}$ .

Table 3.

The root mean square errors of the measured and estimated velocity components

RMSE(·)	Π₁	Π₂	Π₃	Π₄
${\tilde{v}}_{r x} [cm/sec]$	0.1153	0.1172	0.1185	0.5782
${\tilde{v}}_{r y} [cm/sec]$	0.1155	0.1168	0.1183	0.5736
$L^{#} \times {\tilde{ω}}_{r} [c m / s e c]$	0.1157	0.1182	0.1235	0.8058
${\tilde{ω}}_{r} [d e g / s e c]$	0.3159	0.3023	0.4424	28447

Finally, let us assume that a mobile robot is commanded to move forward and rotate at the same time, at a velocity of v_rx = 0 cm/sec, v_ry = 20 cm/sec and ω_r = 1 rad/sec (= 57.3 deg/sec). Again, the same random noise vector n_s is used for both cases of the linear velocity and angular velocity of a mobile robot, for a fair comparison. For the isotropic optical mouse placement Π₁, Figure 11 shows the velocity measurements of the first optical mouse, and the resulting least squares velocity estimation of a mobile robot. Seen from Figures 11(a) and 10(a), the plots of v_1x and v_2x remain the same, but the plots of v_1y and v_2y are shifted by 20 cm/sec. Furthermore, seen from Figures 11(b), all of the plots of ${\tilde{v}}_{r x}, {\tilde{v}}_{r y}, R \times {\tilde{ω}}_{r}$ and ${\tilde{ω}}_{r}$ remain the same. This is because the expression of the Jacobian matrix $\hat{A}$ is affected by the rotation of a mobile robot, but not by the translation of a mobile robot.

Figure 11.

In the case of the isotropic optical mouse placement Π₁, the least squares estimation for a hybrid velocity of a mobile robot: (a) the measured velocity components, v_1x, v_1y, v_2x and v_2y, and (b) the estimated velocity components, ${\tilde{v}}_{r x}$ , ${\tilde{v}}_{r y}$ , $L^{*} \times {\tilde{ω}}_{r}$ , and ${\tilde{ω}}_{r}$ .

9. Some Discussions

In practice, inaccurate optical mouse readings may occur in abnormal situations, typically caused by a sudden change in floor condition or excessive height above the floor [15]. Two representative methods have been proposed to detect the occurrence of abnormal situations and to reduce their effects on mobile robot velocity estimation. First, in [3] and [5], the violation of the kinematic constraint of a pair of optical mice was used to determine the correctness of optical mouse measurements. Second, in [4] and [16], the inconformity of each optical mouse to the consensus reached by all optical mice involved or a pair of optical mice was used to evaluate the reliability of optical mouse measurements. Note that [4] is our previous work. In both methods, the most inaccurate optical mouse measurement is selected and removed from new mobile robot velocity estimation or localization. Furthermore, the ICP (Iterative Closest Point) based localization method was proposed in [14], which is robust to inaccurate optical mouse readings.

In the mobile robot velocity estimation using optical mice, it is essential to maintain a constant height of the optical mice and the floor, as much as possible. As the height increases, the area covered by the camera increases, so that the ratio of the relative displacement counts per real world length is subject to change [15]. To maintain a constant height from the floor, spring mechanisms were suggested to press optical mice to the floor [14, 16]. In [16], two leg springs are placed between a mobile robot and each of the four optical mice, in which small bumps on the floor possibly act as an obstacle to a mobile robot. In [14], two leg springs are placed between a mobile robot and a long arc-bended supporting plate with both of its ends connected to two optical mice by a rotating hinge. It seems that this mechanism can allow optical mice to traverse small bulges and also avoid excessive friction from the floor causing wheel slips [1].

Similar to this paper, the symbolic analysis of the Jacobian matrix and the optimal placement of optical mice were dealt with by Cimino and Pagilla [6]. However, there are two distinctive differences between the two works. First, the symbolic analysis of the Jacobian matrix, in this paper, provides the concise expressions of the eigenvalues, λ₁ and λ₃, in terms of C_x, C_y, and R, each of which has a clear geometrical meaning. With these concise expressions, which are in contrast to the complicated expressions in [6], the geometrical interpretations of both isotropic and singular optical mouse placements can be made, and also the condition for the anisotropic optical mouse placement with maximal isotropy can be readily derived. Second, both works deal with the same problem of optical mouse placement with a different criterion, with possible restrictions on the installation positions of optical mice at the bottom of a mobile robot. In contrast to [6], however, this paper can determine the optimal optical mouse placement, without any assumptions regarding the positions of optical mice.

10. Conclusion

In this paper, we presented the isotropy analysis and optimal placement of optical mice for mobile robot velocity estimation. Positional restrictions on the installation of optical mice at the bottom of a mobile robot were assumed. The main contributions of this paper include 1) the symbolic analysis of the Jacobian matrix, tailored to the isotropy analysis of the optical mouse placement, 2) the identification of the isotropic, anisotropic and singular optical mouse placements along with the corresponding characteristic lengths, and 3) the elaboration of the least squares mobile robot velocity estimation for the isotropic optical mouse placement. The results of this paper can be helpful especially for the development of personal robot mobile platforms with a non-circular base.

Footnotes

11.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2012R1A1A2002175).

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