Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 72
3.8.8 Extraction of Euler Angles from a Given Transformation Matrix
ОглавлениеSuppose a transformation matrix is somehow given as
(3.112)
Then, the Euler angles of a selected sequence can be extracted from by using the procedure explained here. The procedure is explained here for two typical sequences. One of them is the RFB 3‐2‐3 sequence, which is symmetric, and the other one is the 1‐2‐3 sequence, which is asymmetric. However, the same procedure can be used similarly for any other sequence, too.
1 (a) Extraction of the 3‐2‐3 Euler Angles
If the RFB 3‐2‐3 sequence is used, is expressed as
By using the formulas presented in Chapter 2 about the mathematical properties of the rotation matrices, the following set of five scalar equations can be derived from Eq. (3.113) by picking up the appropriate elements of .
From Eq. (3.114), sinφ2 and φ2 can be found as follows with an arbitrary sign variable σ:
(3.119)
(3.120)
(3.121)
If sin φ2 ≠ 0, i.e. if d33 > 0, φ1 and φ3 can be found as follows, respectively, from Eq. Pairs (3.115)–3.116 and (3.117)–3.118 consistently with σ, without introducing any additional sign variable.
(3.122)
(3.123)
(3.124)
(3.125)
(3.126)
(3.127)
1 Selection of the Sign Variable
Based on the solution obtained above for d33 > 0, the following analysis can be made concerning the sign variable σ.
If σ = + 1 leads to , then σ = − 1 leads to , where
(3.128)
Here, and are two independent sign variables, that is, and but they are not necessarily equal. Although and look different, they are actually completely equivalent because they both provide the same transformation matrix as shown below by using the rotation matrix formulas given in Chapter 2.
Equation (3.129) suggests that σ can be selected as σ = + 1 without a significant loss of generality.
1 Singularity Analysis
If sinφ2 = 0, i.e. if d33 = 0, then the 3‐2‐3 sequence becomes singular and the angles φ1 and φ3 cannot be found from Eq. Pairs 3.115,3.116,3.117,3.118 and (3.117)–(3.118), which all reduce to 0 = 0. Such a singularity occurs either if φ2 = 0 or if with . In either case, φ1 and φ3 become indefinite and indistinguishable. So, they cannot be found separately. Nevertheless, their combinations denoted as φ13 = φ1 + φ3 and can still be found depending on whether φ2 = 0 or . The way of finding them is explained below.
If φ2 = 0, Eq. (3.113) can be manipulated as follows:
Equation (3.130) implies that
(3.131)
Hence, φ13 is found as
(3.132)
If , Eq. (3.113) can be manipulated as follows:
Equation (3.133) implies that
(3.134)
Hence, is found as
(3.135)
In order to visualize the singularity of the 3‐2‐3 sequence, the unit vectors of the first and third rotation axes can be expressed as follows in the initial reference frame :
(3.136)
(3.137)
When the singularity occurs with φ2 = 0, becomes
In this singularity, according to Eq. (3.138), the rotations by the angles φ1 and φ3 take place about two axes that have become codirectional. Therefore, only the resultant rotation by the angle φ13 = φ1 + φ3 can be recognized but the angles φ1 and φ3 become obscure and they cannot be distinguished from each other.
When the singularity occurs with , becomes
In this singularity, according to Eq. (3.139), the rotations by the angles φ1 and φ3 take place about two axes that have become oppositely directed. Therefore, only the resultant rotation by the angle can be recognized but the angles φ1 and φ3 become obscure and they cannot be distinguished from each other.
1 (b) Extraction of the 1‐2‐3 Euler Angles
If the RFB 1‐2‐3 sequence is used, is expressed as
Similarly as done above for the 3‐2‐3 Euler angles, the following five scalar equations can be derived from Eq. (3.140) by picking up the appropriate elements of .
From Eq. (3.141), cosφ2 and φ2 can be found as follows with an arbitrary sign variable σ:
(3.146)
(3.147)
In Eq. (3.148), σ2 is different from σ. It is defined as follows if c13 ≠ 0.
(3.149)
If cosφ2 ≠ 0, i.e. if d13 > 0, φ1 and φ3 can be found as follows, respectively, from Eq. Pairs 3.142,3.143,3.144,3.145 and (3.144)–(3.145) consistently with σ, without introducing any additional sign variable.
(3.150)
(3.151)
(3.152)
(3.153)
(3.154)
(3.155)
1 Selection of the Sign Variable
Based on the solution obtained above for d13 > 0, the following analysis can be made concerning the sign variable σ.
If σ = + 1 leads to , then σ = − 1 leads to , where
(3.156)
Here, σ2 = sgn(φ2) as introduced before. As for and , they are two independent sign variables, that is, and but they are not necessarily equal. Although and look different, they are actually completely equivalent because they both provide the same transformation matrix as shown below similarly as done before for the 3‐2‐3 sequence.
According to Eq. (2.87) of Chapter 2 about the three successive half rotations,
Hence, Eq. (3.157) reduces to
Equation (3.158) suggests that σ can again be selected as σ = + 1 without a significant loss of generality.
1 Singularity Analysis
If cosφ2 = 0, i.e. if d13 = 0, the 1‐2‐3 sequence becomes singular and the angles φ1 and φ3 cannot be found from Eq. Pairs, which all reduce to 0 = 0. Such a singularity occurs if with . When it occurs, φ1 and φ3 become indefinite and indistinguishable. So, they cannot be found separately. Nevertheless, their combination denoted as can still be found. The way of finding φ13 is explained below.
In the singularity with , Eq. (3.140) can be manipulated as follows by using the shifting formula given in Chapter 2.
Equation (3.159) implies that
(3.160)
Hence, φ13 is found as
(3.161)
In order to visualize the singularity of the 1‐2‐3 sequence, the unit vectors of the first and third rotation axes can be expressed as follows in the initial reference frame :
(3.162)
(3.163)
When the singularity occurs with , becomes
In this singularity, according to Eq. (3.164), the rotations by the angles φ1 and φ3 take place about two axes that have become parallel, either codirectionally if or oppositely if . Therefore, only the resultant rotation by the angle can be recognized but the angles φ1 and φ3 become obscure and they cannot be distinguished from each other.
