# Degree Of Freedom Analysis Process Essays

## INTRODUCTION

Human walking results from a coordinated sequence of energy generation and absorption (Gordon et al., 1980). During level-ground walking at steady speed, there is an equal balance between positive and negative work production as the body undergoes no net acceleration (assuming negligible external losses). This mechanical work is performed by diverse and distributed physiological tissues, including contributions from both active muscle contractions and passive soft tissue deformations, and affects the kinetic and potential energy of the body. To maintain consistent walking speed, any mechanical energy losses (whether in muscles or in other soft tissues) must be compensated for by net positive work generated by muscles (Kuo et al., 2005). Understanding how, when and where in the body this work is performed is useful for discerning fundamental mechanisms underlying locomotion and can inform applications related to clinical treatment, rehabilitation and assistive technology.

Biomechanical work is often measured at the level of specific joints and body segments, representing the net contributions from underlying muscles, tendons and other tissues. Empirical observations indicate that during walking, substantial positive work is performed about the lower-limb joints (Elftman, 1939; Gordon et al., 1980). For convenience, we use the term ‘joint work’ to describe work performed by muscles, tendons and other structures at/about each joint (e.g. ankle work signifies work performed at/about the ankle joint). The main burst of positive work, termed Push-off, is performed largely by muscles and tendons about the ankle at the end of the Stance phase of gait (Farris and Sawicki, 2012a; Kuo et al., 2005; Winter, 1991) and facilitates economical walking by redirecting the body during step-to-step transitions (Donelan et al., 2002a; Kuo et al., 2005; Ruina et al., 2005).

Joint work estimates, based on inverse dynamics, fail to capture negative work performed by passive soft tissue (DeVita et al., 2007; Zelik and Kuo, 2010) and shoe deformations (Sasaki et al., 2009; Shorten, 1993). For an individual walking on level ground at constant speed, experimental estimates indicate that there is substantially more positive work performed about the lower-limb ankle, knee and hip joints than negative work (DeVita et al., 2007). As positive and negative work must be of equal magnitude for steady-state walking, this difference suggests that the joint-level measures may only capture a portion of the work performed by the body during gait. Additional evidence is based on the comparison of joint work with a separate estimate of the body's center-of-mass (COM) kinetics (Fu et al., 2014; Soo and Donelan, 2010; Zelik and Kuo, 2010, 2012). The mismatch between these estimates indicates that negative work is performed by the body, which cannot be attributed to a specific joint or muscle–tendon source. Also, this mismatch in negative work is larger for obese than for non-obese individuals (Fu et al., 2014), further suggesting that the source may be dissipation by soft tissue deformations in the body.

A similar missing work problem exists for positive work; however, this discrepancy cannot be resolved by invoking soft tissue deformations (as only muscles can perform net positive work). Specifically, if one sums conventional 3 degree-of-freedom (DOF) work measures about the hip, knee and ankle joints (e.g. Zelik and Kuo, 2010) with segment-level contributions from the foot (e.g. Takahashi and Stanhope, 2013), then these estimates fail to account for substantial positive work that is performed by the body (see Materials and methods for complete computational details). When re-analyzing a typical walking data set (Zelik and Kuo, 2010), we found that >30% (∼8 J) of the positive energy change of the body during Push-off, which amounts to ∼25% of the positive energy changes throughout the entire gait cycle, is not captured by conventional joint and foot work estimates (Fig. 1). This is problematic because our measures of work in healthy human gait contribute to our fundamental understanding of locomotion, as well as inform assistive technology development and clinical treatment (e.g. surgical decision-making for children with cerebral palsy; Gage, 1994; Wren et al., 2011).

**Unmeasured positive work.** Push-off kinetics are not explained by conventional joint- and segment-level biomechanical measures, based on our re-analysis of previously published data (Zelik and Kuo, 2010). At 1.4 m s^{−1} there is about 24 J of total positive energy change (Δenergy) during Push-off, which reflects the redirection of the body's center-of-mass (COM) velocity, and also the motion of segmental masses relative to the COM (termed Peripheral energy change). We can compare this total energy change estimate with the mechanical work computed from commonly used joint- and segment-level estimates. We observe that the Push-off work performed by the stance limb joints and foot segment – work performed rotationally (represented by green arrows) about the hip, knee and ankle joints [from 3 degree-of-freedom (DOF) inverse dynamics] and work performed by the foot [a combination of metatarsophalangeal (MTP) joint rotations and other deformations within the foot and shoe] – only sums to about 16 J. Thus, these conventional measures fail to account for 8 J (33%) of the Push-off kinetics.

In this study, we aimed to find and explain the missing positive work; specifically, to determine whether experimental joint and foot work estimates could collectively account for the total mechanical energy change of the body during gait, if estimated with a more sophisticated biomechanical analysis. To accomplish this goal, we extended conventional 3DOF inverse dynamics to a full 6DOF analysis (Buczek et al., 1994; Duncan et al., 1997), and performed a novel ‘Energy-Accounting’ analysis to evaluate biomechanical work and energy. This Energy-Accounting analysis was previously presented in a rudimentary form (Zelik and Kuo, 2012), and builds upon the analytical framework detailed by Aleshinsky (1986). It involves computing several complementary biomechanical estimates (see equations and full details in Materials and methods). Two measures summarize whole-body dynamics: COM and Peripheral rates of energy change, due to motion of the COM and to motion of the limb segments relative to the body's COM, respectively. We refer to the sum of these as the Total rate of energy change of the body. Power estimates were also computed for individual lower-limb joints, based on both 3DOF and 6DOF inverse dynamics. A final power estimate was then computed for the foot.

The specific purpose of this Energy-Accounting analysis was to determine whether and when summed joint and foot segment power (and work) estimates account for the body's Total rate of energy change (and the magnitude of energy change). During human walking, the muscles, tendons and other biological tissues of the lower limb perform work at/about the hip, knee and ankle joints, and in the feet. One of the most common biomechanical estimates is rotational joint power, computed from 3DOF inverse dynamics and denoted as 3DOF power in this study. Recently, foot power estimates computed assuming a deformable segment model have also become more widely used and accepted (Prince et al., 1994; Takahashi et al., 2012). In the absence of a true gold standard (e.g. from a comprehensive array of implantable force and strain gauge measurements), these 3DOF and foot power estimates represent the most commonly used standards for measuring and interpreting contributions from joint- and segment-level sources in the human lower limb. By comparing summed 3DOF+Foot power with the body's Total rate of energy change (of/about the COM), it is then possible to assess the ability of the joint- and segment-level measures to explain whole-body kinetics. In this study, we also extended the conventional 3DOF joint estimates by computing full 6DOF inverse dynamics (Buczek et al., 1994; Duncan et al., 1997), which includes both rotational and translational power terms, and performed a similar comparison of 6DOF+Foot power with Total rate of energy change.

Here, we briefly clarify the work versus energy terminology used in this study. We computed total, COM and Peripheral rate of energy change, then integrated these over time to report energy change (in units of J). In much of the previous biomechanics literature, including our own (Zelik and Kuo, 2012), these integrated values were called work (e.g. COM work, Peripheral work). However, for clarity, it is preferable here to describe these measurements in terms of changes in energy. In particular, Peripheral estimates (see Eqn 3 in Materials and methods) are based solely on changes in kinetic energy, rather than defined by a specific force acting over a displacement (the classical definition of work). COM kinetics are often reported in the literature as the work done to move the body's COM and are indeed calculated from force and displacement (integral of Eqn 1); however, this terminology may be confusing because the (physiological) source of the work is unclear, and the ground reaction forces are not acting directly on the COM. As COM power is equal to the time derivative of the body's kinetic and potential energy (Eqn 2), the integral of COM power also represents the change in energy. In contrast, joint and foot segment power are integrated over time and are reported as mechanical work (also in units of J), as these reflect specific forces/moments acting over specific displacements/angles (e.g. ankle moment acting over measured angular rotation).

In summary, this study poses the question: can the net work performed at or about the lower limb joints and in the foot segment explain the observed changes in the energy state of the body during gait? An alternative phrasing, which may be less clear to some readers but is more consistent with published biomechanics literature, is: can Joint+Segment work account for the Total work performed by the body? Throughout the manuscript, we will use the former phrasing and terminology.

#### List of symbols and abbreviations

- Collision
- phase of gait immediately after footstrike impact, occurring at ∼0–15% of the stride cycle at typical speeds, primarily characterized by a period of negative individual-limb COM power, but also inclusive of positive power transient immediately after footstrike (Fig. 2)
- COM
- center-of-mass
- DOF
- degree-of-freedom
- rate of energy change of the COM
- Peripheral rate of energy change, due to motion of body segments relative to the COM
- Total rate of energy change of the body
- Energy-Accounting analysis
- the name given to our general methodological approach, in which we compare summed joint- and segment-level work estimates with an estimate of the total energy change of the body (depending on the task or animal being studied, the precise formulation of these estimates may vary; see Materials and methods for computations used in this study of human gait)
- ground reaction force under the foot
- inertia
- j
- joint
- Joint+Segment
- in this manuscript, this term refers to contributions from the hip, knee, ankle and foot (although theoretically it could also include additional body joints and segments, if measured)
*m*- mass
- moment
*P*_{3d}- 3DOF joint power
*P*_{6d}- 6DOF joint power
*P*_{com}- COM power
*P*_{foot}- Foot power
- Peripheral
- refers to contributions relative to the body's COM
- Preload
- phase of gait following Rebound, occurring at ∼30–45% of the stride cycle at typical speeds, characterized by negative individual-limb COM power (Fig. 2)
- Push-off
- phase of gait following Preload, occurring at ∼45–65% of the stride cycle at typical speeds, characterized by positive individual-limb COM power (Fig. 2)
- Rebound
- phase of gait following Collision, occurring at ∼15–30% of the stride cycle at typical speeds, characterized by positive individual-limb COM power (Fig. 2)
- s
- segment
- Stance
- period of gait when the ipsilateral foot is on the ground; consists of Collision, Rebound, Preload and Push-off phases of gait
- Swing
- phase of gait following Push-off, occurring at ∼65–100% of the stride cycle at typical speeds, characterized by zero individual-limb COM power as the ipsilateral limb is not in contact with the ground (Fig. 2)
- Total energy change
- sum of COM and Peripheral changes in energy
- COM velocity
- ω
- angular velocity
- 3DOF work
- rotational joint work (based on conventional inverse dynamics)
- 3DOF+Foot work
- sum of rotational joint work and work performed by foot segment deformation
- 6DOF work
- rotational and translational joint work
- 6DOF+Foot work
- sum of rotational and translational joint work and work performed by foot segment deformation

## RESULTS

### Total rate of energy change versus Joint+Segment power

We observed qualitative similarities between the Total rate of energy change, and 3DOF+Foot and 6DOF+Foot power. Each time-varying profile displayed corresponding fluctuations of negative and positive work/energy (Fig. 2A). However, magnitudes varied with phase of gait. 3DOF+Foot and 6DOF+Foot power were in strong agreement with each other during phases that involved principally negative power (Collision, Preload and Swing), but greater differences were observed during periods of positive power (Rebound and Push-off). During Push-off, the 6DOF+Foot power was similar to the Total rate of energy change, but 3DOF+Foot power was smaller in magnitude. During Collision, 3DOF+Foot and 6DOF+Foot power exhibited smaller magnitudes than the Total rate of energy change.

**Mechanical power and rate of energy change.** (A) Summed power and rate of energy change. Three estimates are depicted: 3DOF+Foot power (rotational hip, knee and ankle power+deformable foot power, green dashed line), 6DOF+Foot power (rotational and translational power for all joints and the foot, red solid line), and Total rate of energy change (COM+Peripheral, blue solid line). (B) COM and Peripheral rates of energy change, due to motion of and about the body's COM, respectively, are depicted. (C) Power contributions from individual joints and the foot segment. Conventional 3DOF rotational joint power and full 6DOF joint power are shown. Foot power estimates were only calculated based on a 6DOF deformable body model. Individual-limb results are shown for subjects walking at 1.4 m s^{−1} (*N*=9). Phases of gait – Collision, Rebound, Preload, Push-off and Swing – are depicted by alternating regions of shading.

### 3DOF versus 6DOF joint power

Joint-level differences between 3DOF and 6DOF power were observed mainly at the hip and knee (Fig. 2C). 6DOF hip power was, on average, higher in magnitude than 3DOF estimates, an effect most pronounced during Preload and Push-off. 6DOF knee power displayed a shift towards positive power during Collision, Preload and Push-off. Small differences in ankle power were also observed during Preload and Push-off.

### Push-off work and energy change

Total energy change during Push-off was comparable to 6DOF+Foot work, but not to 3DOF+Foot work (Fig. 3). At 1.4 m s^{−1}, we found 23.7±3.4 J (mean±s.d.) of Total energy change during Push-off (Table 1) and a similar amount of 6DOF+Foot work (22.1±2.5 J, *P*=0.07), but significantly less 3DOF+Foot work (15.8±2.1 J, *P*<0.0001). The 6DOF+Foot work was 6.3 J higher than the 3DOF+Foot work, which accounted for the majority of the missing work at nominal speed (7.9 J). The larger magnitude of the 6DOF+Foot work could be attributed to increased contributions from each lower-limb joint (Fig. 4A), most notably a 55% increase in hip work from 6.0±2.0 to 9.3±1.8 J (3DOF+Foot versus 6DOF+Foot, *P*=0.0002). Knee work during Push-off also changed by ∼30% from −6.7±2.3 to −4.8±2.5 J (*P*=0.0008). The ankle displayed a smaller 5% increase from 22.4±3.7 to 23.6±3.7 J (*P*=0.006).

**Mechanical work and energy change.** 6DOF+Foot work estimates explain Total positive energy changes (Δenergy) during Push-off and positive energy changes across the entire stride cycle; work that is missed by conventional 3DOF+Foot estimates. Results (means and s.d.) are shown for subjects walking at 1.4 m s^{−1} (*N*=9). **P*<0.05. At the bottom, the shaded areas depict regions of integrated positive power and rate of energy change.

**Joint and foot segment work.** On average, 6DOF calculations yielded more positive work than 3DOF estimates at each joint for (A) Push-off work and (B) positive work across the stride. In particular, 6DOF estimates indicate that hip work may, on average, perform >50% more Push-off than conventionally estimated at 1.4 m s^{−1} (*N*=9). Foot power was estimated based on a 6DOF deformable body model and is thus not applicable (N/A) to 3DOF analysis. Data are means and s.d. (**P*<0.05).

**Table 1.**

**Mechanical work and energy change**

### Positive work and energy change over stride

At 1.4 m s^{−1}, the total positive energy change over the stride (39.4±4.4 J; Table 1, Fig. 3) was comparable to the 6DOF+Foot work (40.5±6.5 J, *P*=0.53), whereas 3DOF+Foot work was about 25% less (31.2±6.7 J, *P*=0.0002). On average, 6DOF work magnitudes were larger than 3DOF work at each lower-limb joint: 3.6 J at the hip, 3.2 J at the knee and 1.5 J at the ankle (Fig. 4B), although only ankle and knee differences reached statistical significance. We observed subject-specific hip work differences: five of nine subjects exhibited 6DOF hip work that was >7 J higher than the 3DOF estimate, while the other four subjects exhibited >2 J less hip work at 1.4 m s^{−1}.

### Other phases of gait

Joint+Segment work was in good agreement with Total energy change during other phases of gait, with the exception of Collision. No significant differences were found during Rebound or Preload for Total energy change versus 3DOF+Foot work or Total energy change versus 6DOF+Foot work (*P*>0.08, Table 1). Differences in work during Swing phase were small, on average less than 2 J, although they reached statistical significance. Significant differences were also found during Collision. In terms of the magnitude of negative work, we observed 55% less Joint+Segment Collision work (−5.8±4.1 J of 3DOF+Foot work, −5.7±2.2 J of 6DOF+Foot work) than Total energy change during Collision (−13.1±3.4 J).

### Net work and energy change over stride

The net Total energy change (sum of positive and negative) over the stride was close to zero (<1 J), as expected for steady gait (Table 1). However, Joint+Segment work over the stride was net negative for 3DOF+Foot (approximately −5 J) and net positive for 6DOF+Foot estimates (+4 J).

### Effect of gait speed

Work/energy results were qualitatively consistent across a broad range of speeds, from 0.9 to 2 m s^{−1} (Fig. 5). Total positive energy change across the gait cycle and during Push-off was always significantly higher than 3DOF+Foot work estimates. 6DOF+Foot work consistently provided a better estimate for Total energy change. Total energy change and 6DOF+Foot work were generally not significantly different for Push-off or for positive contributions across the stride, especially at slower speeds. However, a slight degradation in the correspondence of Push-off was observed at higher speeds (>1.4 m s^{−1}). For example, during Push-off, Total energy change and 6DOF+Foot work were in strong agreement at 1.25 m s^{−1} (21.1±3.1 versus 20.5±2.7 J, *P*=0.51), but less so at 1.6 m s^{−1} (28.5±4.7 versus 25.7±3.1 J, *P*=0.01).

**Mechanical work and energy change across walking speed.** Summary measures (means and s.d.) are reported for each phase of gait, and for positive and negative work and Total energy change of a single leg over the entire gait cycle for gait speeds from 0.9 to 2 m s^{−1} (*N*=9).

## DISCUSSION

We integrated various biomechanical analyses to investigate unmeasured positive work during walking. We discovered that the missing work could be explained by extending 3DOF inverse dynamics to 6DOF analysis of the hip, knee, ankle and foot (6DOF+Foot). Our results reaffirm the importance of foot contributions to gait, and revealed that hip Push-off work may be >50% higher than conventionally estimated by 3DOF inverse dynamics. Below, we discuss how these findings advance our biomechanical understanding of human walking, and the implications for experimental and computational research, clinical gait analysis and assistive technology development.

### Accounting for the unmeasured positive work

6DOF+Foot work explained the positive energy changes of/about the body's COM during walking, specifically during gait phases when conventional 3DOF estimates failed to capture much of the body's kinetics (Fig. 3). To our knowledge, this is the first experimental study to reconcile joint- and segment-level positive work generation with the overall energy changes of the body. Previous attempts have demonstrated partial agreement in these estimates, but only during limited portions of the gait cycle (e.g. Winter, 1979). Our findings provide novel and compelling evidence that the 6DOF+Foot approach gives a more accurate and complete estimate of how work is distributed amongst various physiological sources. These improved estimates of biomechanical work advance our empirical knowledge of gait and have potential implications for: (1) assistive technologies (e.g. prostheses, orthoses) that are frequently designed to mimic biological function (Au et al., 2007; Dollar and Herr, 2008; Goldfarb et al., 2013; Lenzi et al., 2013), (2) musculoskeletal simulations of locomotion that are optimized based on empirical biomechanical estimates (Delp et al., 2007; Neptune et al., 2001; Umberger, 2010) and (3) surgical decision-making that relies, in part, on clinical gait analysis and the calculation of joint kinetics to prescribe a surgical plan (e.g. for children with cerebral palsy; Gage, 1994; Wren et al., 2011).

The 6DOF+Foot work estimates were generally in strong agreement with positive changes in Total energy, across subjects and gait speeds (Fig. 5); however, at the highest speeds, we did find that 6DOF+Foot Push-off work corresponded slightly less well (Fig. 5). Differences between Total energy change and 6DOF+Foot work at higher speeds might be due to skin motion artifacts, or larger contributions from the swing limb or from (unmeasured) trunk/arm movement (e.g. a 3 cm vertical excursion of a single arm's COM would contribute about 1 J to raising the body's COM, based on standard anthropometric tables; Winter, 2005). Nevertheless, 6DOF+Foot estimates were consistently found to out-perform 3DOF+Foot estimates across all speeds, with the best correspondence to Total energy change at low to moderate speed (Fig. 5).

### Key scientific implications

6DOF+Foot results indicated that hip muscles and tendons may play a larger role in positive work production than previously estimated (Figs 2 and 4). Much of this hip work is likely due to active muscle contractions, based on the following observations. First, over the gait cycle, we calculated substantially more positive hip work than negative, suggestive of work generated by muscle (although not conclusive because of unknown biarticular muscle–tendon contributions). Second, there was no negative hip work (from either the ipsilateral or contralateral side) immediately preceding the positive Rebound work, which might have been indicative of tendinous energy storage followed by elastic energy return. In contrast, during Push-off, positive hip work might be partially due to elastic tissues (given the preceding negative hip work during Preload). Given the morphology of the hip socket (Cereatti et al., 2010), the intra-joint forces (Ren et al., 2008) and the cartilage thickness (Shepherd and Seedhom, 1999), it is unlikely that substantial work is performed in compression of the hip joint. The underestimate of hip work by 3DOF inverse dynamics may result from methodological limitations (e.g. related to tracking of thigh or pelvic segments). Of all the joints, the hip is perhaps most susceptible to inaccuracies from joint center mislocation, in part due to the inability to place anatomical markers both laterally and medially. Techniques such as functional joint center estimation (Schwartz and Rozumalski, 2005) have been developed to aid joint localization and might improve 3DOF hip work estimates, but this requires future study.

This article is about mechanics. For other fields, see Degrees of freedom.

In physics, the **degree of freedom (DOF)** of a mechanical system is the number of independent parameters that define its configuration. It is the number of parameters that determine the state of a physical system and is important to the analysis of systems of bodies in mechanical engineering, aeronautical engineering, robotics, and structural engineering.

The position of a single railcar (engine) moving along a track has one degree of freedom because the position of the car is defined by the distance along the track. A train of rigid cars connected by hinges to an engine still has only one degree of freedom because the positions of the cars behind the engine are constrained by the shape of the track.

An automobile with highly stiff suspension can be considered to be a rigid body traveling on a plane (a flat, two-dimensional space). This body has three independent degrees of freedom consisting of two components of translation and one angle of rotation. Skidding or drifting is a good example of an automobile's three independent degrees of freedom.

The position and orientation of a rigid body in space is defined by three components of translation and three components of rotation, which means that it has six degrees of freedom.

The exact constraint mechanical design method manages the degrees of freedom to neither underconstrain nor overconstrain a device.^{[1]}

## Motions and dimensions[edit]

The position of an *n*-dimensional rigid body is defined by the rigid transformation, [*T*] = [*A*, *d*], where *d* is an *n*-dimensional translation and *A* is an *n* × *n* rotation matrix, which has *n* translational degrees of freedom and *n*(*n* − 1)/2 rotational degrees of freedom. The number of rotational degrees of freedom comes from the dimension of the rotation group SO(n).

A non-rigid or deformable body may be thought of as a collection of many minute particles (infinite number of DOFs), this is often approximated by a finite DOF system. When motion involving large displacements is the main objective of study (e.g. for analyzing the motion of satellites), a deformable body may be approximated as a rigid body (or even a particle) in order to simplify the analysis.

The degree of freedom of a system can be viewed as the minimum number of coordinates required to specify a configuration. Applying this definition, we have:

- For a single particle in a plane two coordinates define its location so it has two degrees of freedom;
- A single particle in space requires three coordinates so it has three degrees of freedom;
- Two particles in space have a combined six degrees of freedom;
- If two particles in space are constrained to maintain a constant distance from each other, such as in the case of a diatomic molecule, then the six coordinates must satisfy a single constraint equation defined by the distance formula. This reduces the degree of freedom of the system to five, because the distance formula can be used to solve for the remaining coordinate once the other five are specified.

## Six degrees of freedom (6 DoF)[edit]

Main article: Six degrees of freedom

The motion of a ship at sea has the six degrees of freedom of a rigid body, and is described as:^{[2]}

**Translation and rotation:**

- Moving up and down (elevating/heaving);
- Moving left and right (strafing/swaying);
- Moving forward and backward (walking/surging);
- Swivels left and right (yawing);
- Tilts forward and backward (pitching);
- Pivots side to side (rolling).

See also Euler angles

The trajectory of an airplane in flight has three degrees of freedom and its attitude along the trajectory has three degrees of freedom, for a total of six degrees of freedom.

## Mobility formula[edit]

The mobility formula counts the number of parameters that define the configuration of a set of rigid bodies that are constrained by joints connecting these bodies.^{[3]}^{[4]}

Consider a system of *n* rigid bodies moving in space has 6*n* degrees of freedom measured relative to a fixed frame. In order to count the degrees of freedom of this system, include the fixed body in the count of bodies, so that mobility is independent of the choice of the body that forms the fixed frame. Then the degree-of-freedom of the unconstrained system of *N* = *n* + 1 is

because the fixed body has zero degrees of freedom relative to itself.

Joints that connect bodies in this system remove degrees of freedom and reduce mobility. Specifically, hinges and sliders each impose five constraints and therefore remove five degrees of freedom. It is convenient to define the number of constraints *c* that a joint imposes in terms of the joint's freedom *f*, where *c* = 6 − *f*. In the case of a hinge or slider, which are one degree of freedom joints, have *f* = 1 and therefore *c* = 6 − 1 = 5.

The result is that the mobility of a system formed from *n* moving links and *j* joints each with freedom *f*_{i}, *i* = 1, ..., j, is given by

Recall that *N* includes the fixed link.

There are two important special cases: (i) a simple open chain, and (ii) a simple closed chain. A single open chain consists of *n* moving links connected end to end by *n* joints, with one end connected to a ground link. Thus, in this case *N* = *j* + 1 and the mobility of the chain is

For a simple closed chain, *n* moving links are connected end-to-end by *n* + 1 joints such that the two ends are connected to the ground link forming a loop. In this case, we have *N* = *j* and the mobility of the chain is

An example of a simple open chain is a serial robot manipulator. These robotic systems are constructed from a series of links connected by six one degree-of-freedom revolute or prismatic joints, so the system has six degrees of freedom.

An example of a simple closed chain is the RSSR spatial four-bar linkage. The sum of the freedom of these joints is eight, so the mobility of the linkage is two, where one of the degrees of freedom is the rotation of the coupler around the line joining the two S joints.

### Planar and spherical movement[edit]

It is common practice to design the linkage system so that the movement of all of the bodies are constrained to lie on parallel planes, to form what is known as a *planar linkage*. It is also possible to construct the linkage system so that all of the bodies move on concentric spheres, forming a *spherical linkage*. In both cases, the degrees of freedom of the links in each system is now three rather than six, and the constraints imposed by joints are now *c* = 3 − *f*.

In this case, the mobility formula is given by

and the special cases become

- planar or spherical simple open chain,

- planar or spherical simple closed chain,

An example of a planar simple closed chain is the planar four-bar linkage, which is a four-bar loop with four one degree-of-freedom joints and therefore has mobility *M* = 1.

### Systems of bodies[edit]

A system with several bodies would have a combined DOF that is the sum of the DOFs of the bodies, less the internal constraints they may have on relative motion. A mechanism or linkage containing a number of connected rigid bodies may have more than the degrees of freedom for a single rigid body. Here the term *degrees of freedom* is used to describe the number of parameters needed to specify the spatial pose of a linkage.

A specific type of linkage is the open kinematic chain, where a set of rigid links are connected at joints; a joint may provide one DOF (hinge/sliding), or two (cylindrical). Such chains occur commonly in robotics, biomechanics, and for satellites and other space structures. A human arm is considered to have seven DOFs. A shoulder gives pitch, yaw, and roll, an elbow allows for pitch, and a wrist allows for pitch,yaw and roll . Only 3 of those movements would be necessary to move the hand to any point in space, but people would lack the ability to grasp things from different angles or directions. A robot (or object) that has mechanisms to control all 6 physical DOF is said to be holonomic. An object with fewer controllable DOFs than total DOFs is said to be non-holonomic, and an object with more controllable DOFs than total DOFs (such as the human arm) is said to be redundant. Although keep in mind that it is not redundant in the human arm because the two DOFs; wrist and shoulder, that represent the same movement; roll, supply each other since they can't do a full 360. The degree of freedom are like different movements that can be made.

In mobile robotics, a car-like robot can reach any position and orientation in 2-D space, so it needs 3 DOFs to describe its pose, but at any point, you can move it only by a forward motion and a steering angle. So it has two control DOFs and three representational DOFs; i.e. it is non-holonomic. A fixed-wing aircraft, with 3–4 control DOFs (forward motion, roll, pitch, and to a limited extent, yaw) in a 3-D space, is also non-holonomic, as it cannot move directly up/down or left/right.

A summary of formulas and methods for computing the degrees-of-freedom in mechanical systems has been given by Pennestri, Cavacece, and Vita.^{[5]}

## Electrical engineering[edit]

In electrical engineering*degrees of freedom* is often used to describe the number of directions in which a phased arrayantenna can form either beams or nulls. It is equal to one less than the number of elements contained in the array, as one element is used as a reference against which either constructive or destructive interference may be applied using each of the remaining antenna elements. Radar practice and communication link practice, with beam steering being more prevalent for radar applications and null steering being more prevalent for interference suppression in communication links.

## See also[edit]

## References[edit]

**^**http://ocw.mit.edu/courses/mechanical-engineering/2-76-multi-scale-system-design-fall-2004/readings/reading_l3.pdf**^**Summary of ship movementArchived November 25, 2011, at the Wayback Machine.**^**J. J. Uicker, G. R. Pennock, and J. E. Shigley, 2003,**Theory of Machines and Mechanisms,**Oxford University Press, New York.**^**J. M. McCarthy and G. S. Soh,**Geometric Design of Linkages,**2nd Edition, Springer 2010**^**Pennestri E, Cavacece M, Vita L, On the computation of degrees-of-freedom: A didactic perspective, ASME Paper DETC2005-84109

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