Second-Order Geometric Sliding Mode Attitude Observer with Application to Quadrotor on a Test Bench

Abstract

A sliding mode observer design framework is proposed based on the Lie group method of numerical integration on manifolds, and a Second-Order Geometric Sliding Mode Attitude Observer (SOGSMAO) is designed for angular velocity estimation of quadrotor attitude. The algorithm constructs feedback in the angular velocity space and the space of equivalent Lie algebra of unit quaternion space, respectively. It avoids not only the complexity of constructing feedback in unit quaternion space but also the process of mandatory rescaling which is seen to deteriorate the accuracy of the angular velocity estimates during sliding. The performance of SOGSMAO is compared with traditional quaternion based sliding mode observer in which multiplicative quaternion correction is used and the results show that SOGSMAO gains better tracking performance. Then SOGSMAO is realized on a test bed and the effectiveness of the observer algorithm is verified by experimental studies.

1. Introduction

The rotational motion of a rigid body describes the orientation (attitude) of the rigid body with respect to inertial reference. And attitude control is the process of orienting the rigid body in a specified predetermined direction. Attitude control system is an important subsystem in aircrafts, spacecrafts, and satellites and is responsible for pointing and slewing [1–5]. In the case of quadrotor control, translational outer loop is usually realized on the basis of rotational inner loop, as a result of its property of underactuatation [6–9]. Compared with fixed-wing aircraft, rotorcraft has special advantages in the city and indoor environment due to the ability of VTOL (vertical take-off and landing) and hovering flight. As a new kind of rotorcraft, quadrotor has some unique advantages over other traditional rotorcrafts. Firstly, it is more convenient that swash plate and mechanical linkages are not necessary. Secondly, it is much safer in the indoor flying because its propellers are smaller and can be wrapped up. In addition, it has greater thrust-weight ratio and better maneuver performance. Therefore, a great deal of work has been done in quadrotor control.

Most of the attitude control algorithms require the attitude and angular rate. These measurements are obtained from a variety of sensors and estimation techniques. However, the tendency of building smaller satellites and aircrafts limits the number of sensors and the on-board processing capability. So attitude control system requires computationally simple yet effective algorithms without sophisticated measurement instrumentation, particularly gyroscopes [10–14]. Gyroscopes provide direct angular rate measurements but are costly and heavy require significant power consumption, and have high failure tendencies. Lizarralde and Wen use a velocity filter to stabilize attitude without angular velocity [12]. Another solution is to estimate the angular velocity using an extended Kalman filter [15] or a nonlinear observer [13, 14]. In this paper the problem of estimating angular velocities of quadrotor is addressed by nonlinear observers. Estimating angular velocities eliminates the use of gyros and can also confirm the accuracy of gyro readings in case of possible gyro failure and attitude estimates when sensor measurements are not available.

In all kinds of nonlinear observers, sliding mode observer has special advantages. Sliding mode technique is widely known because of its high accuracy and robustness with respect to various internal and external disturbances [16]. Its application is not limited to mechanical systems. Wu and Ho design an SMC for singular systems which can be a natural representation of economic systems, power systems and circuits systems [17]. In addition, there are a wide range of analysis methods such as the Lyapunov method, equivalent control method and dissipativity based method [18, 19]. In the area of observer application, the sliding mode observer has the feature of finite-time convergence, so it is not needed to prove the separation principle when implementing a sliding mode observer in a control system.

In all representations of the attitude, the unit quaternion is a pervasive one [20–22]. Different from minimal attitude representations, such as Euler angles, which have the disadvantages of singularities and complicated trigonometric expressions, the unit quaternion provides a globally nonsingular representation of the attitude. Compared with classical direction cosine matrix, it has only 4 numbers so that the multiplication and propagation rule can be calculated more efficiently. However, unit quaternion is regarded only as an algebraic tool in traditional sliding mode observer algorithms without considering its geometric structure [9, 10]. The classical discrete integral method is used directly on each element of unit quaternion, and the quaternion vector must be normalized in each estimation cycle. In [11, 23] the multiplicative quaternion correction is used instead of the additive quaternion correction, but the discontinuous feedback term is applied in unit quaternion space and it must be normalized too.

Actually, unit quaternion keeps the structure of SU(2), the special unitary group, which is a 2 to 1 cover of SO(3) [21]. The add operation is not a group operation of unit quaternion, so the group structure is not conserved when additive discrete integral method is used. And the normalization is seen to deteriorate the accuracy of the angular velocity estimates during sliding [10]. So a main motivation of this work is to construct an observer design frame from the geometric point of view, such that it is unnecessary to construct sliding mode feedback in unit quaternion space directly or the process of mandatory rescaling can be avoided.

Inspired by the ideological of equivariant mapping in the Lie group method of numerical integration, this work extends the design method of sliding mode observer based on unit quaternion. The kinematic equation in unit quaternion space is transformed to an equivalent differential equation evolving in the Lie algebra space of unit quaternion, and the feedback is constructed in the Lie algebra space. An appealing feature of Lie algebra space is that it is a vector space in which the classical numerical method can be applied in approximately solving the transformed equation. So it is unnecessary to construct sliding mode feedback in unit quaternion space directly in the proposed algorithm. In addition, the process of mandatory rescaling which the traditional methods need in each integration step can be also avoided.

The main contributions of this work are the following.  First, a sliding mode observer design framework is proposed based on the Lie group method of numerical integration on manifolds. In this frame work, the feedback is constructed in vector spaces, so traditional sliding mode observer can be extended easily. Second, a Second-Order Geometric Sliding Mode Attitude Observer (SOGSMAO) is designed for angular velocity estimation of quadrotor attitude. The effectiveness of SOGMAO algorithm is exemplified by simulation and experiment.

This paper is organized as follows. Based on the assumptions of single rigid body and symmetrical structure, a mathematical model of quadrotor on the attitude platform is given in Section 2. Then SOGSMAO is designed using Lie group method of numerical integration and sliding mode technique in Section 3. Section 4 conducts a simulation in which a comparison is made between SOGSMAO and traditional quaternion based sliding mode observer with multiplicative quaternion correction. In Section 5, the performance of the proposed method is evaluated with real experimental data. At last, the conclusion is outlined in Section 6.

2. Attitude Dynamics of Quadrotor

In this section the mathematical model of quadrotor on the test bench is given. The model consists of the kinematics of unit quaternion and the dynamics of single rigid body. In simulation this model is used for providing ideal attitude and angular velocity. And in the process of observer design, the model is a necessity.

2.1. Unit Quaternion Kinematics

A unit quaternion 𝑞 is a complex with four parameters. It has the norm of constant one and can be considered as a point of the unit sphere 𝑆³𝑞=𝑞₀+𝑞₁𝑖+𝑞₂𝑗+𝑞₃𝑞𝑘,₂₀+𝑞²₁+𝑞²₂+𝑞²₃=1. (1)

Let 𝑞 represent its vector part, that is, 𝑞=𝑞₁𝑖+𝑞₂𝑗+𝑞₃𝑘. Then unit quaternion can be written in the following form: 𝑞=𝑞₀,𝑞. (2)

Given a unit quaternion, there is a correspondence between unit quaternion and rotation transformation 𝑅𝑞₀,𝑞=𝑞²₀−𝑞^𝑇𝑞𝐼+2𝑞₀𝑆𝑞+2𝑞𝑞^𝑇, (3) where 𝑆𝑞=0−𝑞₃𝑞₂𝑞₃0−𝑞₁−𝑞₂𝑞₁0. (4)

Noticing that 𝑅(𝑞₀,𝑞)=𝑅(−𝑞₀,−𝑞), so the unit sphere 𝑆³ represented by unit quaternion is a double cover of the attitude space SO(3). The phenomenon is more clearly using the angle form of unit quaternion. Euler’s theorem states that any rotation in space can be achieved through rotating 𝜃 angle around an eigenaxis of 𝑛, then unit quaternion can be written as 𝜃𝑞=cos2𝜃,sin2𝑛,𝜃∈[0,2𝜋]. (5)

It is obvious that rotating 𝜃₀(0≤𝜃₀≤2𝜋) angle around eigenaxis 𝑛 is equivalent to a rotation of 2𝜋−𝜃₀ angle around eigenaxis of −𝑛. In Figure 1, 𝑋 point and 𝑂 point are different by quaternion representation but have the same physical meaning. Actually, 𝑞=(cos(𝜃₀/2),sin(𝜃₀/2)𝑛) and −𝑞=(cos(𝜋−𝜃₀/2),sin(𝜋−𝜃₀/2)𝑛) have the same physical meaning.

Figure 1 

Quaternion representation along some fixed axis.

The conjugate of a unit quaternion is 𝑞^∗=𝑞₀,−𝑞. (6)

Given two unit quaternion 𝑞₁=(𝑞₁₀,𝑞₁) and 𝑞₂=(𝑞₂₀,𝑞₂), the multiplication ⊗ is defined as 𝑞₁⊗𝑞₂=𝑞₁₀𝑞₂₀−𝑞^𝑇₁𝑞₂𝑞₁₀𝑞₂+𝑞₂₀𝑞₁+𝑆𝑞₁𝑞₂. (7)

Unit quaternion multiplication is equivalent to the multiplication of rotation matrix. If 𝑞₁ and 𝑞₂ correspond to rotation matrices 𝑅₁ and 𝑅₂ in SO(3), then there are the following correspondences: 𝑅₁𝑅₂⟷±𝑞₁⊗𝑞₂,𝑅₁𝑣⟷𝑞₁⊗𝑣⊗𝑞^∗₁, (8) where 𝑣=(0,𝑣).

When a rigid body is rotating with angular velocity vector 𝑤, the time derivative of unit quaternion can be given by the unit quaternion propagation rule as follows: 𝑞̇=12𝑞⊗𝑤. (9)

2.2. System Model

Two coordinate frames, the inertial coordinate frame and the body coordinate frame, are employed for modeling the quadrotor attitude system. As seen in Figure 2, the origin of the body coordinate frame 𝑂_𝑏−𝑋_𝑏𝑌_𝑏𝑍_𝑏 is located in the center of the crossbeam of quadrotor, axis 𝑋 points towards propeller 1, axis 𝑍 is perpendicular to the crossbeam plane, and axis three is to form a right-handed coordinate system.

Figure 2 

Coordinate frames of quadrotor.

The quadrotor dynamics are based on two assumptions.(1)Quadrotor is a rigid body.(2)The small errors in manufacturing and installing the vehicle bodies are not considered, so the quadrotor is assumed to be symmetric.

Based on assumption one, the following mathematical model can be derived from Newton-Euler equations and quaternion kinematics:𝐽̇(𝐽𝑤),̇1𝑤=𝑀−𝑤×𝑞=2𝑞₀𝑞𝑞̇𝐼+𝑆𝑤,₀1=−2𝑞^𝑇𝑤, (10a)(10b)where 𝑞=𝑞₀𝑞^𝑇𝑇 represents attitude quaternion of body coordinate frame with respect to inertial coordinate frame, 𝑤 is angular velocity in body coordinate frame, and 𝑀 is the torque acting on the quadrotor.

From the second assumption, the correspondence between four propeller thrusts 𝑓_𝑖(𝑖=1,2,3,4) and the torques 𝑀_𝑖(𝑖=1,2,3) acting on the quadrotor is given as follows: 𝑓𝑀₁𝑀₂𝑀₃=11110𝑑0−𝑑−𝑑0𝑑0−𝑐_𝜏𝑓𝑐_𝜏𝑓−𝑐_𝜏𝑓𝑐_𝜏𝑓𝑓₁𝑓₂𝑓₃𝑓₄, (11) where 𝑓 is the total thrust, 𝑑 is the distance between axis of propeller and the center of crossbeams, and 𝑐_𝜏𝑓 is the propeller torque thrust ratio.

Our model does not include the propeller gyroscopic torque and flapping dynamic term, thus 𝑀 consists of three axis control torque 𝑀₁, 𝑀₂, 𝑀₃𝑀=𝑀₁𝑀₂𝑀₃^𝑇. (12)

Considering the low-speed characteristic of quadrotor, the following simplified equation is used to derive the propeller thrust: 𝑓_𝑖=𝑐_𝑓Ω²_𝑖, (13) where 𝑐_𝑓 is a constant coefficient.

3. Angular Rates Observer Design on Lie Group

Unit quaternion space is a nonlinear manifold 𝑆³. In 𝑆³, classical numerical methods cannot conserve the geometric structure of the original manifold, so a rescaling process must be added which reduces the accuracy of the algorithm and increases the cost of computation. Actually, since the add operation is not a group operation of unit quaternion, it is meaningless in geometric point of view to apply classical numerical methods in 𝑆³.

To solve this problem, a frame work of designing sliding mode observer is proposed using the Lie group method of numerical integration. On the basis of equivariant mapping ideological, the kinematic equation in unit quaternion space is transformed to an equivalent differential equation evolving on the Lie algebra space of unit quaternion, and the feedback is constructed in the Lie algebra space.

3.1. Lie Group Method of Numerical Integration

In this section, only main results of the approach are given, and more detailed material and proofs could be found in [24].

Definition 1 (equivariant map). Let ℳ and 𝒩 be manifolds and let 𝐺 be a Lie group which acts on ℳ by Φ_𝑔∶ℳ→ℳand on 𝒩 by Ψ_𝑔∶𝒩→𝒩. A smooth map 𝖿∶ℳ→𝒩 is called equivariant with respect to these actions if, for all 𝑔∈𝐺𝖿∘Φ_𝑔=Ψ_𝑔∘𝖿. (14)
That is, the following diagram commutes: (15)
From the definition of an action Φ∶𝐺×ℳ→ℳ of 𝐺 on ℳ, we can get an equivariant map Φ_𝑚∶𝐺→ℳ with respect to the left translation action 𝐿_𝑔 of 𝐺 on itself and an action Φ_𝑔 of 𝐺 on ℳΦ_𝑚∘𝐿_𝑔=Φ_𝑔∘Φ_𝑚. (16)
It is known that there is a local coordinate map 𝖿∶𝔤→𝐺 on 𝐺 where 𝔤 is the Lie algebra space of 𝐺, the most typical is exponential map exp. At this point, we need to find an action 𝐵_𝑔 of 𝐺 on 𝔤 such that 𝖿 will be an equivariant map with 𝐵_𝑔 and the left action of 𝐺 on itself 𝖿∘𝐵_𝑔=𝐿_𝑔∘𝖿. (17)
In the case where 𝖿 is the exponential map, 𝐵_𝑔 is nothing else than the well-known Baker-Campbell-Hausdorff (BCH) formula 𝐵_𝑔(𝑢)=log𝑔⋅exp(𝑢), (18) where log is called the logarithm map.
Since composition of two equivariant maps is an equivariant map, we can construct an equivariant map Φ_𝑚∘𝖿 from 𝔤 to ℳ with respect to the action 𝐵_𝑔 on 𝔤 and Φ_𝑔 on ℳ. That is, the following diagram commutes (19)

Definition 2 (relatedness of vector fields). Let 𝜙∶ℳ→𝒩 be a smooth map between manifolds. The vector fields 𝑋 on ℳ and 𝑌 on 𝒩 are called 𝜙-related, denoted by 𝑋∽_𝜙𝑌, if 𝑇𝜙∘𝑋=𝑌∘𝜙.
The definition means that the following diagram commutes: (20)
Let Φ_{exp(𝑡𝜉ℳ)} and Ψ_{exp(𝑡𝜉𝒩)} be two flows on ℳ and 𝒩, respectively, generated by the elements 𝜉∈𝔤 and the action Φ on ℳ and Ψ on 𝒩 of the Lie group 𝐺. The main idea is to solve for a flow on a simpler manifold ℳ so that the effect is the same as if one solved for the flow on 𝒩.
Theorem 3.6 of [25] stated that if and only if 𝜙 is an equivariant map, then the infinitesimal generators of the action with respect to the same element 𝜉∈𝔤 are𝜙-related vector fields. Thus, the infinitesimal generators of the flows of 𝐵_𝑔 and Φ_𝑔 are Φ_𝑚∘𝖿-related, that is, 𝜉_ℳ∘Φ_𝑚∘𝖿=𝑇Φ_𝑚∘𝑇𝖿∘𝜉_𝔤. (21)
That is, the following diagram commutes: (22)
Now, since the infinitesimal generators 𝜉_𝔤 and 𝜉_ℳ on 𝔤 and ℳ respectively is Φ_𝑚∘𝖿-related, the differential equation on ℳ can be translated to a equivalent system on 𝔤 with the selected 𝜉_𝔤. So the differential equation in Lie algebra space 𝔤 can be solved firstly, and the solution of differential equation in corresponding manifold space can be gained.

3.2. Cayley Transform

The kinematics of unit quaternion can be rewritten as 𝑞̇=𝐴(𝑤)𝑞, (23) where 𝐴(𝑤) is the Lie algebra of unit quaternion 𝐴(𝑤)=120−𝑤₃−𝑤₂−𝑤₁𝑤₃0𝑤₃−𝑤₂𝑤₂−𝑤₃0𝑤₁𝑤₁𝑤₂−𝑤₁0. (24)

Then the solution of the differential equation (23) is 𝑞(𝑡)=exp(𝐴(𝑤(𝑡)))⋅𝑞₀. (25)

From the ideological of equivariant mapping, the Lie algebra space of unit quaternion is equivalent to unit quaternion space through equivariant mapping exp(𝑡𝜉). And 𝐴(𝑤(𝑡)) satisfies the following differential equation: 𝐴̇(𝑤(𝑡))=𝑑exp_{𝐴(𝑤(𝑡}⁻¹₎₎(Ω(𝑡)). (26)

In every integral step, the maps of exp(𝑡𝜉) and 𝑑exp_𝑢⁻¹(𝑣) must be solved. But in engineering practice the exact numerical solution of the two maps cannot be computed, and it is usually replaced by some approximate solution. This paper develops integration based on Cayley transform which would entail savings with respect to computational costs because of the less expense in computing the Cayley transform as compared to the exponential map.

The Cayley transform, cay∶𝔤→𝐺, is defined as cay(𝑢)=1+𝑢/2.1−𝑢/2 (27)

The corresponding vector field on the Lie algebra 𝑑cay_𝑢⁻¹ is calculated to be 𝑑cay_𝑢⁻¹(𝑣)1=𝑣−2[]−1𝑢,𝑣4𝑢⋅𝑣⋅𝑢, (28) where [⋅,⋅] denotes the matrix commutator and “⋅” denotes matrix multiplication.

3.3. Second-Order Geometric Sliding Mode Observer Design

The system model is 𝑞̇=12𝑞⊗𝐽̇(𝐽𝑤).𝑤,𝑤=𝑀−𝑤× (29)

Define the error of attitude between estimated attitude and the actual one 𝑞_𝑒≜𝑞^∗⊗𝑞. (30)

When estimated value is equivalent to the actual one, that is, 𝑞=𝑞, 𝑞_𝑒=0, and when 𝑞≠𝑞, 𝑞_𝑒=sin(𝜃/2)𝑛 represent a rotation of 𝜃 angle around axis 𝑛, then 𝑞_𝑒 is considered as feedback variable. Since the attitude of quadrotor is the only measurable output, the sliding manifold is taken as 𝑠=𝑞_𝑒.

Consider the first derivative of the sliding manifold ̇𝑞_𝑒=𝑞₀̇𝑞−𝑞₀𝑞̇̇𝐼−𝑆𝑞−𝑞₀𝑞+𝑞₀−𝑞̇𝑞1𝐼+𝑆=−2𝑞^𝑇𝑤𝑞−𝑞₀̇𝑞̇𝑞+1𝑞+𝑆2𝑞^𝑇𝑤𝑞+𝑞₀̇𝑞̇𝑞1𝑞−𝑆=−2𝑞^𝑇𝑤1𝑞−2𝑞₀𝑞₀𝑞𝑤+1𝐼+𝑆2𝑆𝑞𝑞₀𝑞𝑤+1𝐼+𝑆2𝑞^𝑇𝑤1𝑞+2𝑞₀𝑞₀𝑞𝑤−1𝐼+𝑆2𝑆𝑞𝑞₀𝐼+𝑆𝑞𝑤. (31)

It should be noted that 𝑀 is not included explicitly. But 𝑤 is included in the first derivative of the sliding manifold, so 𝑀 will be included in the second derivative of the sliding manifold, and its sliding order is 2.

The Second-Order Sliding Mode Observer structure in [22] is 𝑥̇₁=𝑥₂+𝑧₁,𝑥̇₂=𝖿𝑡,𝑥₁,𝑥₂,𝑢+𝑧₂, (32) where 𝑧₁=𝜆𝑥₁−𝑥₁^1/2sgn𝑥₁−𝑥₁,𝑧₂=𝛼sgn𝑥₁−𝑥₁. (33)

Then the structure of unit quaternion based SOGSMAO can be designed as ̇1𝑞=2𝑞⊗𝑤−𝑍₁,𝐽̇𝑤=−𝑤×𝐽𝑤−𝑍₂, (34) where 𝑍₁=𝑘₁𝑞_𝑒^1/2⋅SGN𝑞_𝑒,𝑍₂=𝑘₂SGN𝑞_𝑒. (35)

The SGN function is an extended sign function for multidimensional situation. Since the sign function can be seen as the direction function of one dimension, that is, (𝜆)=𝜆sgn|𝜆|𝜆≠00𝜆=0, (36) we extend it to the situation of three dimensions as (𝑋)=𝑋SGN‖𝑋‖[][]𝑋≠000000^𝑇𝑋=[000]. (37)

Based on the Lie group method of numerical integration, the algorithm of SOGSMAO can be transformed to the discrete form as follows: 𝑞(+)𝜉̂𝑞(−),𝜉̂(+)=𝜉̂(−)=cay+ℎ⋅𝑑cay_𝜉⁻¹𝐴𝑤−𝑘₁𝑞_𝑒^1/2⋅SGN𝑞_𝑒,𝑤(+)=𝑤(−)+ℎ⋅𝑀−𝑤×𝐼𝑤−𝑘₂sgn𝑞_𝑒, (38) where ℎ represents time interval between integral steps.

4. Simulation

In this section, the performance of the SOGSMAO is examined in the environment of MATLAB/Simulink. And a comparison is made between SOGSMAO and traditional quaternion based sliding mode observer with multiplicative quaternion correction.

The ideal attitude and angular velocity data is obtained from the model of quadrotor. The inertial parameters used in the simulation are 𝐽=diag8.9429.4587.787∗10⁻³. (39)

The ideal curves are gained from a simulation model in which the initial states are 𝑞(0)=1000^𝑇,𝑤(0)=000. (40)

And the torques acting on the model are 𝑀=0.01∗sin(𝑡)sin(𝑡)0. (41)

The sampled data of system states is shown in Figure 3 (the sample time is 0.02 second).

(a)

(b)

Figure 3 

Ideal system state curves ((a) attitude curves; (b) angular velocity curves).

Observers use attitude data to estimate angular velocity of the quadrotor. The algorithm of SOGSMAO rewritten here is 𝑞(+)𝜉̂𝑞(−),𝜉̂(+)=𝜉̂(−)=cay+ℎ⋅𝑑cay_𝜉⁻¹𝐴𝑤−𝑘₁𝑞_𝑒^1/2⋅SGN𝑞_𝑒,𝑤(+)=𝑤(−)+ℎ⋅𝑀−𝑤×𝐼𝑤−𝑘₂sgn𝑞_𝑒. (42)

And the simulation result of SOGSMAO algorithm is shown in Figure 4.

(a)

(b)

(c)

Figure 4 

The angular velocity estimating result of SOGSMAO (red: ideal angular velocity curves; green: estimated angular velocity curves).

The traditional quaternion based sliding mode observer algorithm with multiplicative quaternion correction can be given as 𝛿𝑞^′_𝐾=𝑘₁𝑞_𝑒^1/2⋅SGN𝑞_𝑒,𝛿𝑞_𝐾=11+𝛿𝑞^′_𝐾1²+𝛿𝑞^′_𝐾2²+𝛿𝑞^′_𝐾3^21/2×1𝛿𝑞^′_𝐾1𝛿𝑞^′_𝐾2𝛿𝑞^′_𝐾3^𝑇,𝑞(+)=𝛿𝑞_𝐾⊗𝑞(−),𝑤(+)=𝑤(−)+ℎ⋅𝑀−𝑤×𝐼𝑤−𝑘₂sgn𝑞_𝑒. (43)

And the simulation result of traditional algorithm is shown in Figure 5.

(a)

(b)

(c)

Figure 5 

The angular velocity estimating results of traditional algorithm (red: ideal angular velocity curves; green: estimated angular velocity curves).

Comparing the results of SOGSMAO algorithm and traditional algorithm for estimating the angular velocity in Figures 4 and 5, it is clear that SOGSMAO algorithm has a better tracking performance while it avoided the process of mandatory rescaling.

Since the sign function SGN(⋅) would cause a high frequency oscillation, a low pass filter must be added for removing chattering while applying the SOGSMAO algorithm. The filtered data of estimated angular velocity is compared with the ideal angular velocity in Figure 6.

(a)

(b)

(c)

Figure 6 

Comparison between the filtered data of SOGSMAO and the ideal angular velocity.

5. Experiment on the Test Bench Quadrotor

A real test was carried out to evaluate the performance of the proposed SOGSMAO. The test configuration is shown in Figure 7. The equipped devices include a posture platform, aircraft body, onboard controller, attitude and heading reference system (AHRS), wireless serial communication interface, remote control and receiver, and a computer.

Figure 7 

Physical devices of the experimental system.

Aircraft body is selected to be the XAircraft X-650 quadrotor which is made by carbon fiber material, equipped with four brushless DC motor and motor speed control device. The empty weight is 800 g, and maximum single propeller thrust is 4N. Attitude and Heading Reference System is the Xsens MTi inertial measurement unit (IMU) which can provide 3 axes drift-less orientation and 3 axes calibrated angular velocity. Its maximum update rate is 100 Hz with 0.5 degree attitude measure accuracy, 1 degree heading measure accuracy, and 300 degrees per second maximum measurable angular velocity. Remote control and receiver are WFLY FT06-A with 5-way continuous pulse channel and a switching channel. TI’s TMS320F2809 chip with 100 MHz core frequency is the core of onboard controller.

The connection diagram of the experimental devices is shown in Figure 8. In every control cycle (20 ms) F2809 chip is interrupted to receive orientation and angular velocity from IMU, then calculate the control pulse width and transfer it to motor speed control devices. Outside the control loop, F2809 is interrupted to turn parameters and send a set of state data to the computer every 100 ms. The remote control instruction is received in the same interrupter program.

Figure 8 

Signal connection diagram.

In the experiment, the sampled data of attitude and angular velocity is gained through wireless serial port with a sample time of 0.02 second (Figures 9 and 10).

(a)

(b)

(c)

(d)

Figure 9 

Quadrotor attitude curves.

(a)

(b)

(c)

Figure 10 

Quadrotor angular velocity curves.

The filtered data of estimated angular velocity is compared with the actual angular velocity in Figure 11. The results show good tracking performance of the observer, which can meet the requirements of control applications in computational burden.

(a)

(b)

(c)

Figure 11 

Comparison between the filtered data of SOGSMAO and the actual angular velocity.

6. Conclusions

The problem of estimating angular velocities of quadrotor is addressed with only attitude measurement in this work. Different from other quaternion based sliding mode observers in which the unit quaternion is considered as an algebraic tool, the proposed sliding mode observer uses the unit quaternion as a nonlinear manifold. Based on the ideological of equivariant mapping, the kinematic equation in unit quaternion space is transformed to an equivalent differential equation evolving on the Lie algebra space of unit quaternion, and the feedback is constructed in the Lie algebra space. The designed algorithm avoids not only the complex sliding mode feedback in unit quaternion space but also the process of mandatory rescaling which the traditional methods need in each integration step. The simulation and experimental results show that the algorithm of Second-Order Geometric Sliding Mode Observer is effective, and can meet the requirements of control applications in computational burden.

The proposed observer design framework has a wide range of applications. A variety of nonlinear observers can be designed based on the framework. In addition, the proposed observer design framework cannot only design the angular rates observer but also design disturbance observer for feedforward compensation.

Acknowledgments

The authors would like to thank Mr. Wanli Li in NUDT for providing advices and Dr. Tao Shen and Baoquan Song in NUDT for the help of constructing the test bench.