CN111930154B - Nonsingular terminal sliding mode control method and device of MEMS gyroscope - Google Patents
Nonsingular terminal sliding mode control method and device of MEMS gyroscope Download PDFInfo
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Abstract
The application discloses a nonsingular terminal sliding mode control method and device of a MEMS gyroscope, which are used for improving the anti-interference capability and the environment adaptability of a MEMS gyroscope driving control system. The method comprises the following steps: constructing a MEMS gyroscope dynamics model considering external interference and influence of a changing working environment; constructing an extended state equation of the MEMS gyroscope dynamics model; constructing a self-adaptive harmonic disturbance observer according to the extended state equation, and designing a frequency update law; constructing a nonsingular terminal sliding mode controller; and driving the MEMS gyroscope dynamics model by adopting the self-adaptive harmonic disturbance observer, the frequency updating law and the nonsingular terminal sliding mode controller.
Description
Technical Field
The invention relates to the field of intelligent instruments and meters, in particular to a nonsingular terminal sliding mode control method and device of a MEMS gyroscope.
Background
Currently, MEMS (Micro-Electro-Mechanical System, microelectromechanical systems) gyroscopes are widely used in the field of measuring angular velocities of robots, consumer electronics, wearable devices. However, since the MEMS gyroscope circuit signal is very weak, the driving control performance thereof is inevitably deteriorated upon being affected by external disturbance or a varying operating environment. Therefore, it is highly desirable to improve the anti-jamming capability (especially the common harmonic interference estimation capability) and the environmental adaptation capability of its drive control system.
Disclosure of Invention
In view of the above, the present invention provides a non-singular terminal sliding mode control method and apparatus for a MEMS gyroscope to improve the anti-interference capability and the environmental adaptability of a MEMS gyroscope driving control system.
A nonsingular terminal sliding mode control method of a MEMS gyroscope comprises the following steps:
constructing a MEMS gyroscope dynamics model considering external interference and influence of a changing working environment;
constructing an extended state equation of the MEMS gyroscope dynamics model;
constructing a self-adaptive harmonic disturbance observer according to the extended state equation, and designing a frequency update law;
constructing a nonsingular terminal sliding mode controller;
and driving the MEMS gyroscope dynamics model by adopting the self-adaptive harmonic disturbance observer, the frequency updating law and the nonsingular terminal sliding mode controller.
Optionally, the constructing a MEMS gyroscope dynamics model that accounts for external disturbances and varying operating environment effects includes:
the MEMS gyro dynamics model considering external interference is as follows:
in the formula (1), x and y are the displacement of the MEMS gyroscope detection mass block along the driving shaft and the detection shaft respectively,for the first derivative of x>Is the second derivative of x>For the first derivative of y, +.>Is the second derivative of y, c xx And c yy For damping coefficient, k xx And k yy Is the rigidity coefficient, k x3 And k y3 Is a nonlinear coefficient, c xy And c yx To damp the coupling coefficient, k xy And k yx For rigidity coupling coefficient, Ω is gyro input angular velocity, k d1 And k d2 As disturbance parameter, k d1 ＞0，k d2 ＞0，d 1 And d 2 External disturbances on the drive shaft and the detection shaft, respectively, u 1 And u 2 Control inputs on the drive shaft and the detection shaft, respectively;
considering the impact of changing working environments, there are: and->Wherein-> And->Are all kinetic constant parameters, deltac xx 、Δc yy 、Δk xx 、Δk yy 、Δc xy 、Δc yx 、Δk xy And Deltak yx All are uncertain parameters which change along with the environment; the external disturbance is expressed asWherein->And->Is interference of constant value->Andis harmonic interference, a 1 、a 2 For unknown amplitude omega 1 、ω 2 For unknown frequency +.> Is an unknown phase;
definition of the definition And->Then the equation (1) is modified to:
in the formula (2), the amino acid sequence of the compound,ΔK 1 and DeltaK 2 For an uncertain parameter matrix that varies with the environment, < +.>Is the first derivative of θ, +.>Is the second derivative of θ.
Optionally, the constructing an extended state equation of the MEMS gyroscope dynamics model includes:
definition z 1 ＝θ，z 3 ＝D h ，/>z 5 =r, then the extended state equation of formula (2) is:
in the formula (3), the amino acid sequence of the compound,is z i First derivative of>For the first derivative of R->
Optionally, the constructing an adaptive harmonic disturbance observer according to the extended state equation and designing a frequency update law include:
the adaptive harmonic disturbance observer is as follows:
wherein,,is z i Estimate of->Is->First derivative of> And->Omega respectively 1 And omega 2 Estimate of alpha i > 0 is the observer gain to be designed;
design frequency update law
Wherein z is 1j ,Z respectively 1 ,/>And Γ is the j-th element of (2) j The value > 0 is the parameter to be designed,is->Is a first derivative of (a).
Optionally, the constructing a nonsingular terminal sliding mode controller includes:
define tracking error as
e＝θ-θ d (6)
Wherein θ is d Is a reference signal;
defining a nonsingular terminal slip plane as
Wherein s= [ s ] 1 s 2 ] T ，e＝[e 1 e 2 ] T ，For the parameter matrix to be designed, the Hurwitz condition is satisfied, < ->And 1 < r 1 < 2 and 1 < r 2 ＜2，r 1 And r 2 Are all parameters to be designed, s 1 Sum s 2 S components on the drive and sense axes, e 1 And e 2 E components on the drive and sense axes, beta 1 And beta 2 β component on the drive axis and the detection axis, respectively, +.>Is the first derivative of e>Is e 1 First derivative of>Is e 2 Is the first derivative of (a);
constructing a nonsingular terminal sliding mode controller of an MEMS gyroscope as
Wherein,,and γ= [ γ ] 1 γ 2 ] T ，0＜γ 1 < 1 and 0 < gamma 2 < 1 is the parameter to be designed, < ->Sum mu 2 > 0 is the parameter to be designed, ">Is theta d Is a second derivative of (c).
Optionally, the driving the MEMS gyroscope dynamics model with the adaptive harmonic disturbance observer, the frequency update law, and the nonsingular terminal sliding mode controller includes:
the MEMS gyroscope dynamics model shown in equation (1) is driven with an adaptive harmonic disturbance observer shown in equation (4), a frequency update law shown in equation (5), and a non-singular terminal sliding mode controller shown in equation (8).
A nonsingular terminal sliding mode control device of a MEMS gyroscope, comprising a processor and a memory, wherein the memory comprises:
the MEMS gyro dynamics model construction unit is used for constructing an MEMS gyro dynamics model considering external interference and the influence of a changing working environment;
the extended state equation construction unit is used for constructing an extended state equation of the MEMS gyroscope dynamics model;
the self-adaptive harmonic disturbance observer construction unit is used for constructing a self-adaptive harmonic disturbance observer according to the extended state equation and designing a frequency update law;
the non-singular terminal sliding mode controller construction unit is used for constructing a non-singular terminal sliding mode controller;
the driving unit is used for driving the MEMS gyroscope dynamics model by adopting the self-adaptive harmonic disturbance observer, the frequency updating law and the nonsingular terminal sliding mode controller;
the MEMS gyroscope dynamics model construction unit, the extended state equation construction unit, the adaptive harmonic disturbance observer construction unit, the nonsingular terminal sliding mode controller construction unit and the driving unit are all stored in a memory as program units, and the processor executes the program units stored in the memory to realize corresponding functions.
According to the technical scheme, the external harmonic interference with unknown dynamics, unknown frequency, unknown amplitude and unknown phase caused by the changing working environment is regarded as state quantity, and an extended state equation of the dynamics of the MEMS gyroscope is introduced; then, the invention designs the self-adaptive harmonic disturbance observer based on the extended state equation, and self-adaptively updates disturbance frequency by using disturbance observation errors, thereby realizing accurate estimation of external harmonic disturbance with unknown frequency, amplitude and phase and unknown dynamics caused by changing working environment, and improving the anti-interference capability and environment adaptability of the MEMS gyroscope drive control system; meanwhile, aiming at the singular problem of a sliding mode function, the invention introduces nonsingular terminal sliding mode control, and selects 1 < r 1 ＜2，1＜r 2 < 2, then when s 1 →0、s 2 When in 0, not only the singular problem which is easy to occur in the sliding mode control is avoided, but also the limited time convergence of the MEMS gyroscope driving control tracking error is realized.
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In order to more clearly illustrate the embodiments of the invention or the technical solutions in the prior art, the drawings that are required in the embodiments or the description of the prior art will be briefly described, it being obvious that the drawings in the following description are only some embodiments of the invention, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of a nonsingular terminal sliding mode control method of a MEMS gyroscope disclosed in the embodiment of the invention;
fig. 2 is a schematic structural diagram of a nonsingular terminal sliding mode control device of a MEMS gyroscope according to an embodiment of the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Referring to fig. 1, the embodiment of the invention discloses a nonsingular terminal sliding mode control method of a MEMS gyroscope, which comprises the following steps:
step S01: and constructing a MEMS gyroscope dynamics model considering external interference and influence of a changing working environment. The specific process is as follows:
the MEMS gyro dynamics model considering external interference is as follows:
in formula (1), x and y are the MEMS gyroscope proof mass along the drive axis and y, respectivelyThe displacement of the shaft is detected and,for the first derivative of x>Is the second derivative of x>For the first derivative of y, +.>Is the second derivative of y, c xx And c yy For damping coefficient, k xx And k yy For rigidity coefficient->And->Is a nonlinear coefficient, c xy And c yx To damp the coupling coefficient, k xy And k yx For rigidity coupling coefficient, Ω is gyro input angular velocity, k d1 And k d2 As disturbance parameter, k d1 ＞0，k d2 ＞0，d 1 And d 2 External disturbances on the drive shaft and the detection shaft, respectively, u 1 And u 2 Control inputs on the drive shaft and the sense shaft, respectively. For example, the designer may choose k d1 =10 and k d2 ＝10。
Considering the impact of changing working environments, there are: and->Wherein-> And->Are all kinetic constant parameters, deltac xx 、Δc yy 、Δk xx 、Δk yy 、Δc xy 、Δc yx 、Δk xy And Deltak yx Are all uncertain parameters which change with the environment. The external disturbance is expressed asWherein->And->Is interference of constant value->Andis harmonic interference, a 1 、a 2 For unknown amplitude omega 1 、ω 2 For unknown frequency +.> Is an unknown phase.
Definition of the definition And->Then the equation (1) is modified to:
in the formula (2), the amino acid sequence of the compound,ΔK 1 and DeltaK 2 For an uncertain parameter matrix that varies with the environment, < +.>Is the first derivative of θ, +.>Is the second derivative of θ. The formula (2), namely the formula (1), is a MEMS gyroscope dynamics model taking external interference and changing working environment influence into consideration.
For example, the designer may choose to
Step S02: and constructing an extended state equation of the MEMS gyroscope dynamics model. The specific process is as follows:
definition z 1 ＝θ，z 3 ＝D h ，/>z 5 =r, then the extended state equation of formula (2) is:
in the formula (3), the amino acid sequence of the compound,is z i First derivative of>For the first derivative of R->
Step S03: and constructing a self-adaptive harmonic disturbance observer according to the extended state equation, and designing a frequency updating law. The specific process is as follows:
the adaptive harmonic disturbance observer is as follows:
wherein,,is z i Estimate of->Is->First derivative of> And->Omega respectively 1 And omega 2 Estimate of alpha i > 0 is the observer gain to be designed. For example, the designer may choose α i ＝1000。
Design frequency update law
Wherein z is 1j ,Z respectively 1 ,/>And Γ is the j-th element of (2) j The value > 0 is the parameter to be designed,is->Is a first derivative of (a). For example, the designer may choose Γ j ＝2。
Step S04: and constructing a nonsingular terminal sliding mode controller. The specific process is as follows:
define tracking error as
e＝θ-θ d (6)
Wherein θ is d Is a reference signal.
For example, the designer may choose θ d ＝[6.2sin(4.71t+π/3)5sin(5.11t-π/6)] T 。
Defining a nonsingular terminal slip plane as
Wherein s= [ s ] 1 s 2 ] T ，e＝[e 1 e 2 ] T ，For the parameter matrix to be designed, the Hurwitz condition is satisfied, < ->And 1 < r 1 < 2 and 1 < r 2 ＜2，r 1 And r 2 All are parameters to be designed. For example, the designer may choose +.>r 1 ＝1.2，r 2 ＝1.2。s 1 Sum s 2 S components on the drive and sense axes, e 1 And e 2 E components on the drive and sense axes, beta 1 And beta 2 β component on the drive axis and the detection axis, respectively, +.>Is the first derivative of e>Is e 1 First derivative of>Is e 2 Is a first derivative of (a).
The nonsingular terminal sliding mode controller for constructing the MEMS gyroscope is as follows
Wherein,,and γ= [ γ ] 1 γ 2 ] T ，0＜γ 1 < 1 and 0 < gamma 2 < 1 is the parameter to be designed, < ->Sum mu 2 > 0 is the parameter to be designed, ">Is theta d Is a second derivative of (c). For example, the designer may choose γ 1 ＝0.5，γ 2 ＝0.5，μ 1 ＝120，μ 2 ＝110。
Step S05: and driving the MEMS gyroscope dynamics model by adopting the self-adaptive harmonic disturbance observer, the frequency updating law and the nonsingular terminal sliding mode controller. The specific process is as follows:
the MEMS gyroscope dynamic model shown in the formula (1) is driven by adopting the self-adaptive harmonic disturbance observer shown in the formula (4), the frequency update law shown in the formula (5) and the nonsingular terminal sliding mode controller shown in the formula (8), so that the gyroscope high-precision driving control is realized, and meanwhile, the unknown dynamics caused by harmonic disturbance and a changing working environment are effectively estimated.
According to the embodiment of the invention, external harmonic interference with unknown dynamics, unknown frequency, unknown amplitude and unknown phase caused by changing working environment is regarded as state quantity, and an extended state equation of MEMS gyroscope dynamics is introduced; then, the embodiment of the invention designs the self-adaptive harmonic disturbance observer based on the extended state equation, and adaptively updates disturbance frequency by using disturbance observation errors, thereby realizing accurate estimation of external harmonic disturbance with unknown frequency, amplitude and phase and unknown dynamics caused by changing working environment, improving the anti-interference capability and the environment adaptation capability of the MEMS gyroscope drive control system, and further improving the drive control performance of the MEMS gyroscope; meanwhile, aiming at the singular problem of the sliding mode function, the embodiment of the invention introduces nonsingular terminal sliding mode control, and selects 1 < r 1 ＜2，1＜r 2 < 2, then when s 1 →0、s 2 When in 0, not only the singular problem which is easy to occur in the sliding mode control is avoided, but also the limited time convergence of the MEMS gyroscope driving control tracking error is realized.
Corresponding to the method embodiment, the embodiment of the invention also discloses a nonsingular terminal sliding mode control device of the MEMS gyroscope, as shown in fig. 2, the nonsingular terminal sliding mode control device of the MEMS gyroscope comprises a processor and a memory, and the memory comprises:
the MEMS gyro dynamics model construction unit 100 is used for constructing a MEMS gyro dynamics model taking the external interference and the influence of a changing working environment into consideration;
an extended state equation construction unit 200, configured to construct an extended state equation of the MEMS gyroscope dynamics model;
an adaptive harmonic disturbance observer construction unit 300, configured to construct an adaptive harmonic disturbance observer according to the extended state equation, and design a frequency update law;
a non-singular terminal sliding mode controller construction unit 400 for constructing a non-singular terminal sliding mode controller;
and a driving unit 500 for driving the MEMS gyroscope dynamics model using the adaptive harmonic disturbance observer, the frequency update law, and the nonsingular terminal sliding mode controller.
The MEMS gyro dynamics model building unit 100, the extended state equation building unit 200, the adaptive harmonic disturbance observer building unit 300, the nonsingular terminal sliding mode controller building unit 400, and the driving unit 500 are all stored as program units in a memory, and the processor executes the program units stored in the memory to realize corresponding functions.
The processor includes a kernel, and the kernel fetches the corresponding program unit from the memory. The kernel can be provided with one or more than one kernel, and the purposes of improving the anti-interference capability and the environment adaptability of the MEMS gyroscope drive control system are achieved by adjusting kernel parameters.
The memory may include volatile memory, random Access Memory (RAM), and/or nonvolatile memory, such as Read Only Memory (ROM) or flash memory (flash RAM), among other forms in computer readable media, the memory including at least one memory chip.
For the nonsingular terminal sliding mode control device of the MEMS gyroscope disclosed by the embodiment of the invention, the nonsingular terminal sliding mode control device corresponds to the nonsingular terminal sliding mode control method of the MEMS gyroscope disclosed by the embodiment, so that the description is simpler, and relevant parts refer to relevant description of the method.
The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the embodiments of the invention. Thus, the present embodiments are not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein. The invention is not described in detail in the general knowledge of a person skilled in the art.
Claims (6)
1. The nonsingular terminal sliding mode control method of the MEMS gyroscope is characterized by comprising the following steps of:
constructing a MEMS gyroscope dynamics model considering external interference and influence of a changing working environment;
constructing an extended state equation of the MEMS gyroscope dynamics model;
constructing a self-adaptive harmonic disturbance observer according to the extended state equation, and designing a frequency update law;
constructing a nonsingular terminal sliding mode controller;
driving the MEMS gyroscope dynamics model by adopting the self-adaptive harmonic disturbance observer, the frequency updating law and the nonsingular terminal sliding mode controller;
the construction of the MEMS gyroscope dynamics model taking external interference and changing working environment influence into consideration comprises the following steps:
the MEMS gyro dynamics model considering external interference is as follows:
in the formula (1), x and y are the displacement of the MEMS gyroscope detection mass block along the driving shaft and the detection shaft respectively,for the first derivative of x>Is the second derivative of x>For the first derivative of y, +.>Is the second derivative of y, c xx And c yy For damping coefficient, k xx And k yy For rigidity coefficient->And->Is a nonlinear coefficient, c xy And c yx To damp the coupling coefficient, k xy And k yx For rigidity coupling coefficient, Ω is gyro input angular velocity, k d1 And k d2 As disturbance parameter, k d1 ＞0，k d2 ＞0，d 1 And d 2 External disturbances on the drive shaft and the detection shaft, respectively, u 1 And u 2 Control inputs on the drive shaft and the detection shaft, respectively;
considering the impact of changing working environments, there are: and->Wherein-> And->Are all kinetic constant parameters, deltac xx 、Δc yy 、Δk xx 、Δk yy 、Δc xy 、Δc yx 、Δk xy And Deltak yx All are uncertain parameters which change along with the environment; the external disturbance is expressed asWherein->And->Is interference of constant value->Andis harmonic interference, a 1 、a 2 For unknown amplitude omega 1 、ω 2 For unknown frequency +.> Is an unknown phase;
definition of the definition Then the equation (1) is modified to:
in the formula (2), the amino acid sequence of the compound,ΔK 1 and DeltaK 2 For an uncertain parameter matrix that varies with the environment, < +.>Is the first derivative of θ, +.>Is the second derivative of θ.
2. The method of claim 1, wherein the constructing the extended state equation of the MEMS gyroscope dynamics model comprises:
definition z 1 ＝θ，z 3 ＝D h ，/>z 5 =r, then the extended state equation of formula (2) is:
in the formula (3), the amino acid sequence of the compound,is z i First derivative of>For the first derivative of R->
3. The method for nonsingular terminal sliding mode control of MEMS gyroscope according to claim 2, wherein the constructing an adaptive harmonic disturbance observer according to the extended state equation and designing a frequency update law comprise:
the adaptive harmonic disturbance observer is as follows:
wherein,,is z i Estimate of->Is->First derivative of> And->Omega respectively 1 And omega 2 Estimate of alpha i > 0 is the observer gain to be designed;
design frequency update law
Wherein,,respectively->And Γ is the j-th element of (2) j > 0 is the parameter to be designed, +.>Is thatIs a first derivative of (a).
4. The method for non-singular terminal sliding mode control of a MEMS gyroscope according to claim 3, wherein said constructing a non-singular terminal sliding mode controller comprises:
define tracking error as
e＝θ-θ d (6)
Wherein θ is d Is a reference signal;
defining a nonsingular terminal slip plane as
Wherein s= [ s ] 1 s 2 ] T ，e＝[e 1 e 2 ] T ，For the parameter matrix to be designed, the Hurwitz condition is satisfied,and 1 < r 1 < 2 and 1 < r 2 ＜2，r 1 And r 2 Are all parameters to be designed, s 1 Sum s 2 S components on the drive and sense axes, e 1 And e 2 E components on the drive and sense axes, beta 1 And beta 2 β component on the drive axis and the detection axis, respectively, +.>Is the first derivative of e>Is e 1 First derivative of>Is e 2 Is the first derivative of (a);
constructing a nonsingular terminal sliding mode controller of an MEMS gyroscope as
Wherein,,and γ= [ γ ] 1 γ 2 ] T ，0＜γ 1 < 1 and 0 < gamma 2 < 1 is the parameter to be designed, < ->Sum mu 2 > 0 is the parameter to be designed, ">Is theta d Is a second derivative of (c).
5. The method of claim 4, wherein driving the MEMS gyroscope dynamics model using the adaptive harmonic disturbance observer, the frequency update law, and the non-singular terminal sliding mode controller comprises:
the MEMS gyroscope dynamics model shown in equation (1) is driven with an adaptive harmonic disturbance observer shown in equation (4), a frequency update law shown in equation (5), and a non-singular terminal sliding mode controller shown in equation (8).
6. A nonsingular terminal sliding mode control device of a MEMS gyroscope, comprising a processor and a memory, wherein the memory comprises:
the MEMS gyro dynamics model construction unit is used for constructing an MEMS gyro dynamics model considering external interference and the influence of a changing working environment;
the extended state equation construction unit is used for constructing an extended state equation of the MEMS gyroscope dynamics model;
the self-adaptive harmonic disturbance observer construction unit is used for constructing a self-adaptive harmonic disturbance observer according to the extended state equation and designing a frequency update law;
the non-singular terminal sliding mode controller construction unit is used for constructing a non-singular terminal sliding mode controller;
the driving unit is used for driving the MEMS gyroscope dynamics model by adopting the self-adaptive harmonic disturbance observer, the frequency updating law and the nonsingular terminal sliding mode controller;
the MEMS gyroscope dynamics model construction unit, the extended state equation construction unit, the self-adaptive harmonic disturbance observer construction unit, the nonsingular terminal sliding mode controller construction unit and the driving unit are all stored in a memory as program units, and a processor executes the program units stored in the memory to realize corresponding functions;
the construction of the MEMS gyroscope dynamics model taking external interference and changing working environment influence into consideration comprises the following steps:
the MEMS gyro dynamics model considering external interference is as follows:
in the formula (1), x and y are the displacement of the MEMS gyroscope detection mass block along the driving shaft and the detection shaft respectively,for the first derivative of x>Is the second derivative of x>For the first derivative of y, +.>Is the second derivative of y, c xx And c yy For damping coefficient, k xx And k yy For rigidity coefficient->And->Is a nonlinear coefficient, c xy And c yx To damp the coupling coefficient, k xy And k yx For rigidity coupling coefficient, Ω is gyro input angular velocity, k d1 And k d2 As disturbance parameter, k d1 ＞0，k d2 ＞0，d 1 And d 2 External disturbances on the drive shaft and the detection shaft, respectively, u 1 And u 2 Control inputs on the drive shaft and the detection shaft, respectively;
considering the impact of changing working environments, there are: and->Wherein-> And->Are all kinetic constant parameters, deltac xx 、Δc yy 、Δk xx 、Δk yy 、Δc xy 、Δc yx 、Δk xy And Deltak yx All are uncertain parameters which change along with the environment; the external disturbance is expressed asWherein->And->Is interference of constant value->Andis harmonic interference, a 1 、a 2 For unknown amplitude omega 1 、ω 2 For unknown frequency +.> Is an unknown phase;
definition of the definition And->Then the equation (1) is modified to:
in the formula (2), the amino acid sequence of the compound,ΔK 1 and DeltaK 2 For an uncertain parameter matrix that varies with the environment, < +.>Is the first derivative of θ, +.>Is the second derivative of θ.
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